:: N-Tuples and Cartesian Products for n=5
:: by Micha{\l} Muzalewski and Wojciech Skaba
::
:: Received October 13, 1990
:: Copyright (c) 1990-2011 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies SUBSET_1, XBOOLE_0, ZFMISC_1, MCART_1, TARSKI, MCART_2;
notations TARSKI, XBOOLE_0, ZFMISC_1, SUBSET_1, MCART_1;
constructors TARSKI, MCART_1;
registrations XBOOLE_0;
requirements SUBSET, BOOLE;
theorems TARSKI, ZFMISC_1, MCART_1, XBOOLE_0, XBOOLE_1;
schemes XBOOLE_0;
begin
reserve v,z for set;
reserve x,x1,x2,x3,x4,x5,x6,x7,x8,x9 for set;
reserve y,y1,y2,y3,y4,y5,y6,y7,y8,y9 for set;
reserve X,X1,X2,X3,X4,X5,X6,X7,X8,X9 for set;
reserve Y,Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB,YC,YD,YE,YF for set;
reserve Z,Z1,Z2,Z3,Z4,Z5,Z6,Z7,Z8,Z9,ZA,ZB,ZC,ZD,ZE,ZF for set;
reserve xx1 for Element of X1;
reserve xx2 for Element of X2;
reserve xx3 for Element of X3;
reserve xx4 for Element of X4;
reserve xx5 for Element of X5;
reserve xx6 for Element of X6;
reserve xx7 for Element of X7;
reserve xx8 for Element of X8;
reserve xx9 for Element of X9;
reserve A1 for Subset of X1,
A2 for Subset of X2,
A3 for Subset of X3,
A4 for Subset of X4,
A5 for Subset of X5,
A6 for Subset of X6,
A7 for Subset of X7,
A8 for Subset of X8,
A9 for Subset of X9;
Lm1: X1 <> {} & X2 <> {} implies for x being Element of [:X1,X2:] ex xx1 being
(Element of X1), xx2 being Element of X2 st x = [xx1,xx2]
proof
assume
A1: X1 <> {} & X2 <> {};
then
A2: [:X1,X2:] <> {} by ZFMISC_1:90;
let x be Element of [:X1,X2:];
reconsider xx2 = x`2 as Element of X2 by A2,MCART_1:10;
reconsider xx1 = x`1 as Element of X1 by A2,MCART_1:10;
take xx1,xx2;
thus thesis by A1,MCART_1:22;
end;
Lm2: X1 <> {} & X2 <> {} & X3 <> {} implies for x being Element of [:X1,X2,X3
:] ex xx1,xx2,xx3 st x = [xx1,xx2,xx3]
proof
assume that
A1: X1 <> {} & X2 <> {} and
A2: X3 <> {};
let x be Element of [:X1,X2,X3:];
reconsider x9=x as Element of [:[:X1,X2:],X3:] by ZFMISC_1:def 3;
[:X1,X2:] <> {} by A1,ZFMISC_1:90;
then consider x12 being (Element of [:X1,X2:]), xx3 such that
A3: x9 = [x12,xx3] by A2,Lm1;
consider xx1,xx2 such that
A4: x12 = [xx1,xx2] by A1,Lm1;
take xx1,xx2,xx3;
thus thesis by A3,A4,MCART_1:def 3;
end;
Lm3: X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} implies for x being Element of
[:X1,X2,X3,X4:] ex xx1,xx2,xx3,xx4 st x = [xx1,xx2,xx3,xx4]
proof
assume that
A1: X1 <> {} & X2 <> {} & X3 <> {} and
A2: X4 <> {};
let x be Element of [:X1,X2,X3,X4:];
reconsider x9=x as Element of [:[:X1,X2,X3:],X4:] by ZFMISC_1:def 4;
[:X1,X2,X3:] <> {} by A1,MCART_1:31;
then consider x123 being (Element of [:X1,X2,X3:]), xx4 such that
A3: x9 = [x123,xx4] by A2,Lm1;
consider xx1,xx2,xx3 such that
A4: x123 = [xx1,xx2,xx3] by A1,Lm2;
take xx1,xx2,xx3,xx4;
thus thesis by A3,A4,MCART_1:def 4;
end;
theorem
X <> {} implies ex Y st Y in X & for Y1,Y2,Y3,Y4,Y5,Y6 st Y1 in Y2 &
Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y holds Y1 misses X
proof
defpred P[set] means ex Y1,Y2,Y3,Y4,Y5 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 &
Y4 in Y5 & Y5 in $1 & Y1 meets X;
consider Z1 such that
A1: for Y holds Y in Z1 iff Y in union X & P[Y] from XBOOLE_0:sch 1;
defpred U[set] means $1 meets X;
defpred T[set] means ex Y1 st Y1 in $1 & Y1 meets X;
defpred S[set] means ex Y1,Y2 st Y1 in Y2 & Y2 in $1 & Y1 meets X;
defpred R[set] means ex Y1,Y2,Y3 st Y1 in Y2 & Y2 in Y3 & Y3 in $1 & Y1
meets X;
defpred Q[set] means ex Y1,Y2,Y3,Y4 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4
in $1 & Y1 meets X;
consider Z2 such that
A2: for Y holds Y in Z2 iff Y in union union X & Q[Y] from XBOOLE_0:sch
1;
consider Z6 such that
A3: for Y holds Y in Z6 iff Y in union union union union union union X &
U[Y] from XBOOLE_0:sch 1;
consider Z3 such that
A4: for Y holds Y in Z3 iff Y in union union union X & R[Y] from
XBOOLE_0:sch 1;
consider Z5 such that
A5: for Y holds Y in Z5 iff Y in union union union union union X & T[Y]
from XBOOLE_0:sch 1;
consider Z4 such that
A6: for Y holds Y in Z4 iff Y in union union union union X & S[Y] from
XBOOLE_0:sch 1;
set V = X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6;
assume X <> {};
then consider Y such that
A7: Y in V and
A8: Y misses V by MCART_1:1;
A9: V = X \/ (Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5) \/ Z6 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6) by XBOOLE_1:4;
A10: now
assume
A11: Y in Z1;
then consider Y1,Y2,Y3,Y4,Y5 such that
A12: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 and
A13: Y5 in Y and
A14: Y1 meets X by A1;
Y in union X by A1,A11;
then Y5 in union union X by A13,TARSKI:def 4;
then Y5 in Z2 by A2,A12,A14;
then Y5 in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
then Y meets X \/ Z1 \/ Z2 by A13,XBOOLE_0:3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_1:70;
hence contradiction by A8,XBOOLE_1:70;
end;
A15: now
assume
A16: Y in Z2;
then consider Y1,Y2,Y3,Y4 such that
A17: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 and
A18: Y4 in Y and
A19: Y1 meets X by A2;
Y in union union X by A2,A16;
then Y4 in union union union X by A18,TARSKI:def 4;
then Y4 in Z3 by A4,A17,A19;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 3;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y4 in V by XBOOLE_0:def 3;
hence contradiction by A8,A18,XBOOLE_0:3;
end;
A20: now
assume
A21: Y in Z3;
then consider Y1,Y2,Y3 such that
A22: Y1 in Y2 & Y2 in Y3 and
A23: Y3 in Y and
A24: Y1 meets X by A4;
Y in union union union X by A4,A21;
then Y3 in union union union union X by A23,TARSKI:def 4;
then Y3 in Z4 by A6,A22,A24;
then Y3 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then Y3 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y3 in V by XBOOLE_0:def 3;
hence contradiction by A8,A23,XBOOLE_0:3;
end;
A25: now
assume
A26: Y in Z4;
then consider Y1,Y2 such that
A27: Y1 in Y2 and
A28: Y2 in Y and
A29: Y1 meets X by A6;
Y in union union union union X by A6,A26;
then Y2 in union union union union union X by A28,TARSKI:def 4;
then Y2 in Z5 by A5,A27,A29;
then Y2 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y2 in V by XBOOLE_0:def 3;
hence contradiction by A8,A28,XBOOLE_0:3;
end;
A30: now
assume
A31: Y in Z5;
then consider Y1 such that
A32: Y1 in Y and
A33: Y1 meets X by A5;
Y in union union union union union X by A5,A31;
then Y1 in union union union union union union X by A32,TARSKI:def 4;
then Y1 in Z6 by A3,A33;
then Y1 in V by XBOOLE_0:def 3;
hence contradiction by A8,A32,XBOOLE_0:3;
end;
assume
A34: not thesis;
now
assume
A35: Y in X;
then consider Y1,Y2,Y3,Y4,Y5,Y6 such that
A36: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 and
A37: Y6 in Y and
A38: not Y1 misses X by A34;
Y6 in union X by A35,A37,TARSKI:def 4;
then Y6 in Z1 by A1,A36,A38;
then Y6 in X \/ Z1 by XBOOLE_0:def 3;
then Y6 in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 3;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by A37,XBOOLE_0:3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_1:70;
hence contradiction by A8,XBOOLE_1:70;
end;
then Y in Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by A9,A7,XBOOLE_0:def 3;
then Y in Z1 \/ (Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5) \/ Z6 by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6) by XBOOLE_1:4;
then Y in Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by A10,XBOOLE_0:def 3;
then Y in Z2 \/ (Z3 \/ Z4) \/ Z5 \/ Z6 by XBOOLE_1:4;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5) \/ Z6 by XBOOLE_1:4;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5 \/ Z6) by XBOOLE_1:4;
then Y in Z3 \/ Z4 \/ Z5 \/ Z6 by A15,XBOOLE_0:def 3;
then Y in Z3 \/ (Z4 \/ Z5) \/ Z6 by XBOOLE_1:4;
then Y in Z3 \/ (Z4 \/ Z5 \/ Z6) by XBOOLE_1:4;
then Y in Z4 \/ Z5 \/ Z6 by A20,XBOOLE_0:def 3;
then Y in Z4 \/ (Z5 \/ Z6) by XBOOLE_1:4;
then Y in Z5 \/ Z6 by A25,XBOOLE_0:def 3;
then Y in Z6 by A30,XBOOLE_0:def 3;
then Y meets X by A3;
hence contradiction by A9,A8,XBOOLE_1:70;
end;
theorem Th2:
X <> {} implies ex Y st Y in X & for Y1,Y2,Y3,Y4,Y5,Y6,Y7 st Y1
in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y holds Y1
misses X
proof
defpred P[set] means ex Y1,Y2,Y3,Y4,Y5,Y6 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4
& Y4 in Y5 & Y5 in Y6 & Y6 in $1 & Y1 meets X;
consider Z1 such that
A1: for Y holds Y in Z1 iff Y in union X & P[Y] from XBOOLE_0:sch 1;
defpred V[set] means $1 meets X;
defpred U[set] means ex Y1 st Y1 in $1 & Y1 meets X;
defpred T[set] means ex Y1,Y2 st Y1 in Y2 & Y2 in $1 & Y1 meets X;
defpred S[set] means ex Y1,Y2,Y3 st Y1 in Y2 & Y2 in Y3 & Y3 in $1 & Y1
meets X;
defpred R[set] means ex Y1,Y2,Y3,Y4 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4
in $1 & Y1 meets X;
defpred Q[set] means ex Y1,Y2,Y3,Y4,Y5 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 &
Y4 in Y5 & Y5 in $1 & Y1 meets X;
consider Z2 such that
A2: for Y holds Y in Z2 iff Y in union union X & Q[Y] from XBOOLE_0:sch
1;
consider Z7 such that
A3: for Y holds Y in Z7 iff Y in union union union union union union
union X & V[Y] from XBOOLE_0:sch 1;
consider Z3 such that
A4: for Y holds Y in Z3 iff Y in union union union X & R[Y] from
XBOOLE_0:sch 1;
consider Z6 such that
A5: for Y holds Y in Z6 iff Y in union union union union union union X &
U[Y] from XBOOLE_0:sch 1;
consider Z5 such that
A6: for Y holds Y in Z5 iff Y in union union union union union X & T[Y]
from XBOOLE_0:sch 1;
consider Z4 such that
A7: for Y holds Y in Z4 iff Y in union union union union X & S[Y] from
XBOOLE_0:sch 1;
set V = X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7;
assume X <> {};
then consider Y such that
A8: Y in V and
A9: Y misses V by MCART_1:1;
A10: V = X \/ (Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5) \/ Z6 \/ Z7 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6) \/ Z7 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7) by XBOOLE_1:4;
A11: now
assume
A12: Y in Z1;
then consider Y1,Y2,Y3,Y4,Y5,Y6 such that
A13: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 and
A14: Y6 in Y and
A15: Y1 meets X by A1;
Y in union X by A1,A12;
then Y6 in union union X by A14,TARSKI:def 4;
then Y6 in Z2 by A2,A13,A15;
then Y6 in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
then Y meets X \/ Z1 \/ Z2 by A14,XBOOLE_0:3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:70;
hence contradiction by A9,XBOOLE_1:70;
end;
A16: now
assume
A17: Y in Z2;
then consider Y1,Y2,Y3,Y4,Y5 such that
A18: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 and
A19: Y5 in Y and
A20: Y1 meets X by A2;
Y in union union X by A2,A17;
then Y5 in union union union X by A19,TARSKI:def 4;
then Y5 in Z3 by A4,A18,A20;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 3;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then Y5 in V by XBOOLE_0:def 3;
hence contradiction by A9,A19,XBOOLE_0:3;
end;
assume
A21: not thesis;
now
assume
A22: Y in X;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7 such that
A23: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 and
A24: Y7 in Y and
A25: not Y1 misses X by A21;
Y7 in union X by A22,A24,TARSKI:def 4;
then Y7 in Z1 by A1,A23,A25;
then Y7 in X \/ Z1 by XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by A24,XBOOLE_0:3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:70;
hence contradiction by A9,XBOOLE_1:70;
end;
then Y in Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by A10,A8,XBOOLE_0:def 3;
then Y in Z1 \/ (Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5) \/ Z6 \/ Z7 by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6) \/ Z7 by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7) by XBOOLE_1:4;
then Y in Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by A11,XBOOLE_0:def 3;
then Y in Z2 \/ (Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:4;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5) \/ Z6 \/ Z7 by XBOOLE_1:4;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5 \/ Z6) \/ Z7 by XBOOLE_1:4;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7) by XBOOLE_1:4;
then Y in Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by A16,XBOOLE_0:def 3;
then Y in Z3 \/ (Z4 \/ Z5) \/ Z6 \/ Z7 by XBOOLE_1:4;
then Y in Z3 \/ (Z4 \/ Z5 \/ Z6) \/ Z7 by XBOOLE_1:4;
then
A26: Y in Z3 \/ (Z4 \/ Z5 \/ Z6 \/ Z7) by XBOOLE_1:4;
A27: now
assume
A28: Y in Z4;
then consider Y1,Y2,Y3 such that
A29: Y1 in Y2 & Y2 in Y3 and
A30: Y3 in Y and
A31: Y1 meets X by A7;
Y in union union union union X by A7,A28;
then Y3 in union union union union union X by A30,TARSKI:def 4;
then Y3 in Z5 by A6,A29,A31;
then Y3 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y3 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then Y3 in V by XBOOLE_0:def 3;
hence contradiction by A9,A30,XBOOLE_0:3;
end;
A32: now
assume
A33: Y in Z5;
then consider Y1,Y2 such that
A34: Y1 in Y2 and
A35: Y2 in Y and
A36: Y1 meets X by A6;
Y in union union union union union X by A6,A33;
then Y2 in union union union union union union X by A35,TARSKI:def 4;
then Y2 in Z6 by A5,A34,A36;
then Y2 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then Y2 in V by XBOOLE_0:def 3;
hence contradiction by A9,A35,XBOOLE_0:3;
end;
A37: now
assume
A38: Y in Z6;
then consider Y1 such that
A39: Y1 in Y and
A40: Y1 meets X by A5;
Y in union union union union union union X by A5,A38;
then Y1 in union union union union union union union X by A39,TARSKI:def 4;
then Y1 in Z7 by A3,A40;
then Y1 in V by XBOOLE_0:def 3;
hence contradiction by A9,A39,XBOOLE_0:3;
end;
now
assume
A41: Y in Z3;
then consider Y1,Y2,Y3,Y4 such that
A42: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 and
A43: Y4 in Y and
A44: Y1 meets X by A4;
Y in union union union X by A4,A41;
then Y4 in union union union union X by A43,TARSKI:def 4;
then Y4 in Z4 by A7,A42,A44;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then Y4 in V by XBOOLE_0:def 3;
hence contradiction by A9,A43,XBOOLE_0:3;
end;
then Y in Z4 \/ Z5 \/ Z6 \/ Z7 by A26,XBOOLE_0:def 3;
then Y in Z4 \/ (Z5 \/ Z6) \/ Z7 by XBOOLE_1:4;
then Y in Z4 \/ (Z5 \/ Z6 \/ Z7) by XBOOLE_1:4;
then Y in Z5 \/ Z6 \/ Z7 by A27,XBOOLE_0:def 3;
then Y in Z5 \/ (Z6 \/ Z7) by XBOOLE_1:4;
then Y in Z6 \/ Z7 by A32,XBOOLE_0:def 3;
then Y in Z7 by A37,XBOOLE_0:def 3;
then Y meets X by A3;
hence contradiction by A10,A9,XBOOLE_1:70;
end;
::
:: Tuples for n=5
::
definition
let x1,x2,x3,x4,x5;
func [x1,x2,x3,x4,x5] equals
[[x1,x2,x3,x4],x5];
correctness;
end;
theorem Th3:
[x1,x2,x3,x4,x5] = [[[[x1,x2],x3],x4],x5]
proof
thus [x1,x2,x3,x4,x5] = [[[x1,x2,x3],x4],x5] by MCART_1:def 4
.= [[[[x1,x2],x3],x4],x5] by MCART_1:def 3;
end;
theorem
[x1,x2,x3,x4,x5] = [[x1,x2,x3],x4,x5]
proof
thus [x1,x2,x3,x4,x5] = [[[[x1,x2],x3],x4],x5] by Th3
.= [[[x1,x2],x3],x4,x5] by MCART_1:def 3
.= [[x1,x2,x3],x4,x5] by MCART_1:def 3;
end;
theorem Th6:
[x1,x2,x3,x4,x5] = [[x1,x2],x3,x4,x5]
proof
thus [x1,x2,x3,x4,x5] = [[[[x1,x2],x3],x4],x5] by Th3
.= [[[x1,x2],x3],x4,x5] by MCART_1:def 3
.= [[x1,x2],x3,x4,x5] by MCART_1:28;
end;
theorem Th7:
[x1,x2,x3,x4,x5] = [y1,y2,y3,y4,y5] implies x1 = y1 & x2 = y2 &
x3 = y3 & x4 = y4 & x5 = y5
proof
assume
A1: [x1,x2,x3,x4,x5] = [y1,y2,y3,y4,y5];
then [x1,x2,x3,x4] = [y1,y2,y3,y4] by ZFMISC_1:27;
hence thesis by A1,MCART_1:29,ZFMISC_1:27;
end;
theorem Th8:
X <> {} implies ex x st x in X & not ex x1,x2,x3,x4,x5 st (x1 in
X or x2 in X) & x = [x1,x2,x3,x4,x5]
proof
assume X <> {};
then consider Y such that
A1: Y in X and
A2: for Y1,Y2,Y3,Y4,Y5,Y6,Y7 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in
Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y holds Y1 misses X by Th2;
take x = Y;
thus x in X by A1;
given x1,x2,x3,x4,x5 such that
A3: x1 in X or x2 in X and
A4: x = [x1,x2,x3,x4,x5];
set Y1 = { x1, x2 }, Y2 = { Y1, {x1} }, Y3 = { Y2, x3 }, Y4 = { Y3, {Y2} },
Y5 = { Y4, x4 }, Y6 = { Y5, {Y4} }, Y7 = { Y6, x5 };
A5: Y3 in Y4 & Y4 in Y5 by TARSKI:def 2;
Y = [[[[x1,x2],x3],x4],x5] by A4,Th3
.= [[[ Y2,x3],x4],x5 ] by TARSKI:def 5
.= [[ Y4,x4],x5 ] by TARSKI:def 5
.= [ Y6,x5 ] by TARSKI:def 5
.= { Y7, { Y6 } } by TARSKI:def 5;
then
A6: Y7 in Y by TARSKI:def 2;
A7: Y5 in Y6 & Y6 in Y7 by TARSKI:def 2;
x1 in Y1 & x2 in Y1 by TARSKI:def 2;
then
A8: not Y1 misses X by A3,XBOOLE_0:3;
Y1 in Y2 & Y2 in Y3 by TARSKI:def 2;
hence contradiction by A2,A8,A5,A7,A6;
end;
::
:: Cartesian products of five sets
::
definition
let X1,X2,X3,X4,X5;
func [:X1,X2,X3,X4,X5:] -> set equals
[:[: X1,X2,X3,X4 :],X5 :];
correctness;
end;
theorem Th9:
[:X1,X2,X3,X4,X5:] = [:[:[:[:X1,X2:],X3:],X4:],X5:]
proof
thus [:X1,X2,X3,X4,X5:] = [:[:[:X1,X2,X3:],X4:],X5:] by ZFMISC_1:def 4
.= [:[:[:[:X1,X2:],X3:],X4:],X5:] by ZFMISC_1:def 3;
end;
theorem
[:X1,X2,X3,X4,X5:] = [:[:X1,X2,X3:],X4,X5:]
proof
thus [:X1,X2,X3,X4,X5:] = [:[:[:[:X1,X2:],X3:],X4:],X5:] by Th9
.= [:[:[:X1,X2:],X3:],X4,X5:] by ZFMISC_1:def 3
.= [:[:X1,X2,X3:],X4,X5:] by ZFMISC_1:def 3;
end;
theorem Th12:
[:X1,X2,X3,X4,X5:] = [:[:X1,X2:],X3,X4,X5:]
proof
thus [:X1,X2,X3,X4,X5:] = [:[:[:[:X1,X2:],X3:],X4:],X5:] by Th9
.= [:[:[:X1,X2:],X3:],X4,X5:] by ZFMISC_1:def 3
.= [:[:X1,X2:],X3,X4,X5:] by MCART_1:50;
end;
theorem Th13:
X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} iff [:X1,X2
,X3,X4,X5:] <> {}
proof
X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} iff [:X1,X2,X3,X4:] <> {} by
MCART_1:51;
hence thesis by ZFMISC_1:90;
end;
theorem Th14:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} implies ( [:X1,X2,X3,
X4,X5:] = [:Y1,Y2,Y3,Y4,Y5:] implies X1=Y1 & X2=Y2 & X3=Y3 & X4=Y4 & X5=Y5 )
proof
assume
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{};
then
A2: [:X1,X2,X3,X4:] <> {} by MCART_1:51;
assume
A3: X5<>{};
assume
A4: [:X1,X2,X3,X4,X5:] = [:Y1,Y2,Y3,Y4,Y5:];
then [:X1,X2,X3,X4:] = [:Y1,Y2,Y3,Y4:] by A2,A3,ZFMISC_1:110;
hence thesis by A1,A2,A3,A4,MCART_1:52,ZFMISC_1:110;
end;
theorem
[:X1,X2,X3,X4,X5:]<>{} & [:X1,X2,X3,X4,X5:] = [:Y1,Y2,Y3,Y4,Y5:]
implies X1=Y1 & X2=Y2 & X3=Y3 & X4=Y4 & X5=Y5
proof
assume
A1: [:X1,X2,X3,X4,X5:]<>{};
then
A2: X3<>{} & X4<>{} by Th13;
A3: X5<>{} by A1,Th13;
X1<>{} & X2<>{} by A1,Th13;
hence thesis by A2,A3,Th14;
end;
theorem
[:X,X,X,X,X:] = [:Y,Y,Y,Y,Y:] implies X = Y
proof
assume [:X,X,X,X,X:] = [:Y,Y,Y,Y,Y:];
then X<>{} or Y<>{} implies thesis by Th14;
hence thesis;
end;
theorem Th17:
X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} implies for
x being Element of [:X1,X2,X3,X4,X5:] ex xx1,xx2,xx3,xx4,xx5 st x = [xx1,xx2,
xx3,xx4,xx5]
proof
assume that
A1: X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} and
A2: X5 <> {};
let x be Element of [:X1,X2,X3,X4,X5:];
reconsider x9=x as Element of [:[:X1,X2,X3,X4:],X5:];
[:X1,X2,X3,X4:] <> {} by A1,MCART_1:51;
then consider x1234 being (Element of [:X1,X2,X3,X4:]), xx5 such that
A3: x9 = [x1234,xx5] by A2,Lm1;
consider xx1,xx2,xx3,xx4 such that
A4: x1234 = [xx1,xx2,xx3,xx4] by A1,Lm3;
take xx1,xx2,xx3,xx4,xx5;
thus thesis by A3,A4;
end;
definition
let X1,X2,X3,X4,X5;
assume
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{};
let x be Element of [:X1,X2,X3,X4,X5:];
func x`1 -> Element of X1 means
:Def3:
x = [x1,x2,x3,x4,x5] implies it = x1;
existence
proof
consider xx1,xx2,xx3,xx4,xx5 such that
A2: x = [xx1,xx2,xx3,xx4,xx5] by A1,Th17;
take xx1;
thus thesis by A2,Th7;
end;
uniqueness
proof
let y,z be Element of X1;
assume
A3: x = [x1,x2,x3,x4,x5] implies y = x1;
assume
A4: x = [x1,x2,x3,x4,x5] implies z = x1;
consider xx1,xx2,xx3,xx4,xx5 such that
A5: x = [xx1,xx2,xx3,xx4,xx5] by A1,Th17;
y = xx1 by A5,A3;
hence thesis by A5,A4;
end;
func x`2 -> Element of X2 means
:Def4:
x = [x1,x2,x3,x4,x5] implies it = x2;
existence
proof
consider xx1,xx2,xx3,xx4,xx5 such that
A6: x = [xx1,xx2,xx3,xx4,xx5] by A1,Th17;
take xx2;
thus thesis by A6,Th7;
end;
uniqueness
proof
let y,z be Element of X2;
assume
A7: x = [x1,x2,x3,x4,x5] implies y = x2;
assume
A8: x = [x1,x2,x3,x4,x5] implies z = x2;
consider xx1,xx2,xx3,xx4,xx5 such that
A9: x = [xx1,xx2,xx3,xx4,xx5] by A1,Th17;
y = xx2 by A9,A7;
hence thesis by A9,A8;
end;
func x`3 -> Element of X3 means
:Def5:
x = [x1,x2,x3,x4,x5] implies it = x3;
existence
proof
consider xx1,xx2,xx3,xx4,xx5 such that
A10: x = [xx1,xx2,xx3,xx4,xx5] by A1,Th17;
take xx3;
thus thesis by A10,Th7;
end;
uniqueness
proof
let y,z be Element of X3;
assume
A11: x = [x1,x2,x3,x4,x5] implies y = x3;
assume
A12: x = [x1,x2,x3,x4,x5] implies z = x3;
consider xx1,xx2,xx3,xx4,xx5 such that
A13: x = [xx1,xx2,xx3,xx4,xx5] by A1,Th17;
y = xx3 by A13,A11;
hence thesis by A13,A12;
end;
func x`4 -> Element of X4 means
:Def6:
x = [x1,x2,x3,x4,x5] implies it = x4;
existence
proof
consider xx1,xx2,xx3,xx4,xx5 such that
A14: x = [xx1,xx2,xx3,xx4,xx5] by A1,Th17;
take xx4;
thus thesis by A14,Th7;
end;
uniqueness
proof
let y,z be Element of X4;
assume
A15: x = [x1,x2,x3,x4,x5] implies y = x4;
assume
A16: x = [x1,x2,x3,x4,x5] implies z = x4;
consider xx1,xx2,xx3,xx4,xx5 such that
A17: x = [xx1,xx2,xx3,xx4,xx5] by A1,Th17;
y = xx4 by A17,A15;
hence thesis by A17,A16;
end;
func x`5 -> Element of X5 means
:Def7:
x = [x1,x2,x3,x4,x5] implies it = x5;
existence
proof
consider xx1,xx2,xx3,xx4,xx5 such that
A18: x = [xx1,xx2,xx3,xx4,xx5] by A1,Th17;
take xx5;
thus thesis by A18,Th7;
end;
uniqueness
proof
let y,z be Element of X5;
assume
A19: x = [x1,x2,x3,x4,x5] implies y = x5;
assume
A20: x = [x1,x2,x3,x4,x5] implies z = x5;
consider xx1,xx2,xx3,xx4,xx5 such that
A21: x = [xx1,xx2,xx3,xx4,xx5] by A1,Th17;
y = xx5 by A21,A19;
hence thesis by A21,A20;
end;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} implies for x being Element
of [:X1,X2,X3,X4,X5:] for x1,x2,x3,x4,x5 st x = [x1,x2,x3,x4,x5] holds x`1 = x1
& x`2 = x2 & x`3 = x3 & x`4 = x4 & x`5 = x5 by Def3,Def4,Def5,Def6,Def7;
theorem Th19:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} implies for x being
Element of [:X1,X2,X3,X4,X5:] holds x = [x`1,x`2,x`3,x`4,x`5]
proof
assume
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{};
let x be Element of [:X1,X2,X3,X4,X5:];
consider xx1,xx2,xx3,xx4,xx5 such that
A2: x = [xx1,xx2,xx3,xx4,xx5] by A1,Th17;
thus x = [x`1,xx2,xx3,xx4,xx5] by A1,A2,Def3
.= [x`1,x`2,xx3,xx4,xx5] by A1,A2,Def4
.= [x`1,x`2,x`3,xx4,xx5] by A1,A2,Def5
.= [x`1,x`2,x`3,x`4,xx5] by A1,A2,Def6
.= [x`1,x`2,x`3,x`4,x`5] by A1,A2,Def7;
end;
theorem Th20:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} implies for x being
Element of [:X1,X2,X3,X4,X5:] holds x`1 = (x qua set)`1`1`1`1 & x`2 = (x qua
set)`1`1`1`2 & x`3 = (x qua set)`1`1`2 & x`4 = (x qua set)`1`2 & x`5 = (x qua
set)`2
proof
assume
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{};
let x be Element of [:X1,X2,X3,X4,X5:];
thus x`1 = [ x`1, x`2]`1 by MCART_1:7
.= [[x`1, x`2],x`3]`1`1 by MCART_1:7
.= [x`1, x`2 ,x`3]`1`1 by MCART_1:def 3
.= [[x`1, x`2 ,x`3],x`4]`1`1`1 by MCART_1:7
.= [x`1, x`2 ,x`3 ,x`4]`1`1`1 by MCART_1:def 4
.= [x`1, x`2 ,x`3 ,x`4 , x`5]`1`1`1`1 by MCART_1:7
.= (x qua set)`1`1`1`1 by A1,Th19;
thus x`2 = [x`1, x`2]`2 by MCART_1:7
.= [[x`1, x`2],x`3]`1`2 by MCART_1:7
.= [x`1, x`2, x`3]`1`2 by MCART_1:def 3
.= [[x`1, x`2, x`3],x`4]`1`1`2 by MCART_1:7
.= [x`1, x`2, x`3, x`4]`1`1`2 by MCART_1:def 4
.= [x`1, x`2, x`3, x`4 , x`5]`1`1`1`2 by MCART_1:7
.= (x qua set)`1`1`1`2 by A1,Th19;
thus x`3 = [[x`1, x`2],x`3]`2 by MCART_1:7
.= [ x`1, x`2, x`3]`2 by MCART_1:def 3
.= [[x`1, x`2, x`3],x`4]`1`2 by MCART_1:7
.= [ x`1, x`2, x`3, x`4]`1`2 by MCART_1:def 4
.= [x`1, x`2, x`3, x`4 ,x`5]`1`1`2 by MCART_1:7
.= (x qua set)`1`1`2 by A1,Th19;
thus x`4 = [[x`1,x`2,x`3],x`4]`2 by MCART_1:7
.= [x`1,x`2,x`3, x`4]`2 by MCART_1:def 4
.= [x`1,x`2,x`3, x`4 ,x`5]`1`2 by MCART_1:7
.= (x qua set)`1`2 by A1,Th19;
thus x`5 = [x`1,x`2,x`3,x`4 ,x`5]`2 by MCART_1:7
.= (x qua set)`2 by A1,Th19;
end;
theorem Th21:
X1 c= [:X1,X2,X3,X4,X5:] or X1 c= [:X2,X3,X4,X5,X1:] or X1 c= [:
X3,X4,X5,X1,X2:] or X1 c= [:X4,X5,X1,X2,X3:] or X1 c= [:X5,X1,X2,X3,X4:]
implies X1 = {}
proof
assume that
A1: X1 c= [:X1,X2,X3,X4,X5:] or X1 c= [:X2,X3,X4,X5,X1:] or X1 c= [:X3,
X4,X5,X1,X2:] or X1 c= [:X4,X5,X1,X2,X3:] or X1 c= [:X5,X1,X2,X3,X4:] and
A2: X1 <> {};
A3: [:X1,X2,X3,X4,X5:]<>{} or [:X2,X3,X4,X5,X1:]<>{} or [:X3,X4,X5,X1,X2:]<>
{} or [:X4,X5,X1,X2,X3:]<>{} or [:X5,X1,X2,X3,X4:]<>{} by A1,A2,XBOOLE_1:3;
then
A4: X4<>{} & X5<>{} by Th13;
A5: X2<>{} & X3<>{} by A3,Th13;
now
per cases by A1;
suppose
A6: X1 c= [:X1,X2,X3,X4,X5:];
consider x such that
A7: x in X1 and
A8: for x1,x2,x3,x4,x5 st x1 in X1 or x2 in X1 holds x <> [x1,x2,x3
,x4,x5] by A2,Th8;
reconsider x as Element of [:X1,X2,X3,X4,X5:] by A6,A7;
x = [x`1,x`2,x`3,x`4,x`5] by A2,A5,A4,Th19;
hence contradiction by A2,A8;
end;
suppose
X1 c= [:X2,X3,X4,X5,X1:];
then X1 c= [:[:X2,X3:],X4,X5,X1:] by Th12;
hence thesis by A2,MCART_1:59;
end;
suppose
X1 c= [:X3,X4,X5,X1,X2:];
then X1 c= [:[:X3,X4:],X5,X1,X2:] by Th12;
hence thesis by A2,MCART_1:59;
end;
suppose
X1 c= [:X4,X5,X1,X2,X3:];
then X1 c= [:[:X4,X5:],X1,X2,X3:] by Th12;
hence thesis by A2,MCART_1:59;
end;
suppose
A9: X1 c= [:X5,X1,X2,X3,X4:];
consider x such that
A10: x in X1 and
A11: for x1,x2,x3,x4,x5 st x1 in X1 or x2 in X1 holds x <> [x1,x2,x3
,x4,x5] by A2,Th8;
reconsider x as Element of [:X5,X1,X2,X3,X4:] by A9,A10;
x = [x`1,x`2,x`3,x`4,x`5] by A2,A5,A4,Th19;
hence thesis by A2,A11;
end;
end;
hence contradiction;
end;
theorem
[:X1,X2,X3,X4,X5:] meets [:Y1,Y2,Y3,Y4,Y5:] implies X1 meets Y1 & X2
meets Y2 & X3 meets Y3 & X4 meets Y4 & X5 meets Y5
proof
assume
A1: [:X1,X2,X3,X4,X5:] meets [:Y1,Y2,Y3,Y4,Y5:];
[:[:[:[:X1,X2:],X3:],X4:],X5:] = [:X1,X2,X3,X4,X5:] & [:[:[:[:Y1,Y2:],Y3
:], Y4:],Y5:] = [:Y1,Y2,Y3,Y4,Y5:] by Th9;
then
A2: [:[:[:X1,X2:],X3:],X4:] meets [:[:[:Y1,Y2:],Y3:],Y4:] by A1,ZFMISC_1:104;
then
A3: [:[:X1,X2:],X3:] meets [:[:Y1,Y2:],Y3:] by ZFMISC_1:104;
then [:X1,X2:] meets [:Y1,Y2:] by ZFMISC_1:104;
hence thesis by A1,A2,A3,ZFMISC_1:104;
end;
theorem Th23:
[:{x1},{x2},{x3},{x4},{x5}:] = { [x1,x2,x3,x4,x5] }
proof
thus [:{x1},{x2},{x3},{x4},{x5}:] = [:[:{x1},{x2}:],{x3},{x4},{x5}:] by Th12
.= [:{[x1,x2]},{x3},{x4},{x5}:] by ZFMISC_1:29
.= { [[x1,x2], x3, x4, x5]} by MCART_1:61
.= { [x1,x2,x3,x4,x5] } by Th6;
end;
:: 5 - Tuples
reserve x for Element of [:X1,X2,X3,X4,X5:];
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} implies for x1,x2,x3,x4,x5
st x = [x1,x2,x3,x4,x5] holds x`1 = x1 & x`2 = x2 & x`3 = x3 & x`4 = x4 & x`5 =
x5 by Def3,Def4,Def5,Def6,Def7;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & (for xx1,xx2,xx3,xx4,xx5
st x = [xx1,xx2,xx3,xx4,xx5] holds y1 = xx1) implies y1 =x`1
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} and
A2: for xx1,xx2,xx3,xx4,xx5 st x = [xx1,xx2,xx3,xx4,xx5] holds y1 = xx1;
x = [x`1,x`2,x`3,x`4,x`5] by A1,Th19;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & (for xx1,xx2,xx3,xx4,xx5
st x = [xx1,xx2,xx3,xx4,xx5] holds y2 = xx2) implies y2 =x`2
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} and
A2: for xx1,xx2,xx3,xx4,xx5 st x = [xx1,xx2,xx3,xx4,xx5] holds y2 = xx2;
x = [x`1,x`2,x`3,x`4,x`5] by A1,Th19;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & (for xx1,xx2,xx3,xx4,xx5
st x = [xx1,xx2,xx3,xx4,xx5] holds y3 = xx3) implies y3 =x`3
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} and
A2: for xx1,xx2,xx3,xx4,xx5 st x = [xx1,xx2,xx3,xx4,xx5] holds y3 = xx3;
x = [x`1,x`2,x`3,x`4,x`5] by A1,Th19;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & (for xx1,xx2,xx3,xx4,xx5
st x = [xx1,xx2,xx3,xx4,xx5] holds y4 = xx4) implies y4 =x`4
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} and
A2: for xx1,xx2,xx3,xx4,xx5 st x = [xx1,xx2,xx3,xx4,xx5] holds y4 = xx4;
x = [x`1,x`2,x`3,x`4,x`5] by A1,Th19;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & (for xx1,xx2,xx3,xx4,xx5
st x = [xx1,xx2,xx3,xx4,xx5] holds y5 = xx5) implies y5 =x`5
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} and
A2: for xx1,xx2,xx3,xx4,xx5 st x = [xx1,xx2,xx3,xx4,xx5] holds y5 = xx5;
x = [x`1,x`2,x`3,x`4,x`5] by A1,Th19;
hence thesis by A2;
end;
theorem Th30:
y in [: X1,X2,X3,X4,X5 :] implies ex x1,x2,x3,x4,x5 st x1 in X1
& x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & y = [x1,x2,x3,x4,x5]
proof
assume y in [: X1,X2,X3,X4,X5 :];
then consider x1234, x5 being set such that
A1: x1234 in [:X1,X2,X3,X4:] and
A2: x5 in X5 and
A3: y = [x1234,x5] by ZFMISC_1:def 2;
consider x1, x2, x3, x4 such that
A4: x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 and
A5: x1234 = [x1,x2,x3,x4] by A1,MCART_1:79;
y = [x1,x2,x3,x4,x5] by A3,A5;
hence thesis by A2,A4;
end;
theorem Th31:
[x1,x2,x3,x4,x5] in [: X1,X2,X3,X4,X5 :] iff x1 in X1 & x2 in X2
& x3 in X3 & x4 in X4 & x5 in X5
proof
A1: [x1,x2] in [:X1,X2:] iff x1 in X1 & x2 in X2 by ZFMISC_1:87;
[:X1,X2,X3,X4,X5:] = [:[:X1,X2:],X3,X4,X5:] & [x1,x2,x3,x4,x5] = [[x1,x2
],x3,x4,x5] by Th6,Th12;
hence thesis by A1,MCART_1:80;
end;
theorem
(for y holds y in Z iff ex x1,x2,x3,x4,x5 st x1 in X1 & x2 in X2 & x3
in X3 & x4 in X4 & x5 in X5 & y = [x1,x2,x3,x4,x5]) implies Z = [: X1,X2,X3,X4,
X5 :]
proof
assume
A1: for y holds y in Z iff ex x1,x2,x3,x4,x5 st x1 in X1 & x2 in X2 & x3
in X3 & x4 in X4 & x5 in X5 & y = [x1,x2,x3,x4,x5];
now
let y;
thus y in Z implies y in [:[:X1,X2,X3,X4:],X5:]
proof
assume y in Z;
then consider x1,x2,x3,x4,x5 such that
A2: x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 and
A3: x5 in X5 & y = [x1,x2,x3,x4,x5] by A1;
[x1,x2,x3,x4] in [:X1,X2,X3,X4:] by A2,MCART_1:80;
hence thesis by A3,ZFMISC_1:def 2;
end;
assume y in [:[:X1,X2,X3,X4:],X5:];
then consider x1234,x5 being set such that
A4: x1234 in [:X1,X2,X3,X4:] and
A5: x5 in X5 and
A6: y = [x1234,x5] by ZFMISC_1:def 2;
consider x1,x2,x3,x4 such that
A7: x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 and
A8: x1234 = [x1,x2,x3,x4] by A4,MCART_1:79;
y = [x1,x2,x3,x4,x5] by A6,A8;
hence y in Z by A1,A5,A7;
end;
hence thesis by TARSKI:1;
end;
theorem Th33:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & Y1<>{} & Y2<>{} &
Y3<>{} & Y4<>{} & Y5<>{} implies for x being Element of [:X1,X2,X3,X4,X5:], y
being Element of [:Y1,Y2,Y3,Y4,Y5:] holds x = y implies x`1 = y`1 & x`2 = y`2 &
x`3 = y`3 & x`4 = y`4 & x`5 = y`5
proof
assume that
A1: X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} and
A2: Y1 <> {} & Y2 <> {} & Y3 <> {} & Y4 <> {} & Y5 <> {};
let x be Element of [:X1,X2,X3,X4,X5:];
let y be Element of [:Y1,Y2,Y3,Y4,Y5:];
assume
A3: x = y;
thus x`1 = (x qua set)`1`1`1`1 by A1,Th20
.= y`1 by A2,A3,Th20;
thus x`2 = (x qua set)`1`1`1`2 by A1,Th20
.= y`2 by A2,A3,Th20;
thus x`3 = (x qua set)`1`1`2 by A1,Th20
.= y`3 by A2,A3,Th20;
thus x`4 = (x qua set)`1`2 by A1,Th20
.= y`4 by A2,A3,Th20;
thus x`5 = (x qua set)`2 by A1,Th20
.= y`5 by A2,A3,Th20;
end;
theorem
for x being Element of [:X1,X2,X3,X4,X5:] st x in [:A1,A2,A3,A4,A5:]
holds x`1 in A1 & x`2 in A2 & x`3 in A3 & x`4 in A4 & x`5 in A5
proof
let x be Element of [:X1,X2,X3,X4,X5:];
assume
A1: x in [:A1,A2,A3,A4,A5:];
then reconsider y = x as Element of [:A1,A2,A3,A4,A5:];
A2 <> {} by A1,Th13;
then
A2: y`2 in A2;
A4 <> {} by A1,Th13;
then
A3: y`4 in A4;
A3 <> {} by A1,Th13;
then
A4: y`3 in A3;
A5 <> {} by A1,Th13;
then
A5: y`5 in A5;
A1 <> {} by A1,Th13;
then y`1 in A1;
hence thesis by A2,A4,A3,A5,Th33;
end;
theorem Th35:
X1 c= Y1 & X2 c= Y2 & X3 c= Y3 & X4 c= Y4 & X5 c= Y5 implies [:
X1,X2,X3,X4,X5:] c= [:Y1,Y2,Y3,Y4,Y5:]
proof
assume X1 c= Y1 & X2 c= Y2 & X3 c= Y3 & X4 c= Y4;
then
A1: [:X1,X2,X3,X4:] c= [:Y1,Y2,Y3,Y4:] by MCART_1:84;
assume X5 c= Y5;
hence thesis by A1,ZFMISC_1:96;
end;
definition
let X1,X2,X3,X4,X5,A1,A2,A3,A4,A5;
redefine func [:A1,A2,A3,A4,A5:] -> Subset of [:X1,X2,X3,X4,X5:];
coherence by Th35;
end;
theorem
X1 <> {} & X2 <> {} implies for xx being Element of [:X1,X2:] ex xx1
being Element of X1, xx2 being Element of X2 st xx = [xx1,xx2] by Lm1;
theorem
X1 <> {} & X2 <> {} & X3 <> {} implies for xx being Element of [:X1,X2
,X3:] ex xx1,xx2,xx3 st xx = [xx1,xx2,xx3] by Lm2;
theorem
X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} implies for xx being Element
of [:X1,X2,X3,X4:] ex xx1,xx2,xx3,xx4 st xx = [xx1,xx2,xx3,xx4] by Lm3;
begin
theorem
X <> {} implies ex Y st Y in X & for Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8 st Y1 in
Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y
holds Y1 misses X
proof
defpred P1[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7 st Y1 in Y2 & Y2 in Y3 & Y3 in
Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in $1 & Y1 meets X;
consider Z1 such that
A1: for Y holds Y in Z1 iff Y in union X & P1[Y] from XBOOLE_0:sch 1;
defpred P8[set] means $1 meets X;
defpred P7[set] means ex Y1 st Y1 in $1 & Y1 meets X;
defpred P6[set] means ex Y1,Y2 st Y1 in Y2 & Y2 in $1 & Y1 meets X;
defpred P5[set] means ex Y1,Y2,Y3 st Y1 in Y2 & Y2 in Y3 & Y3 in $1 & Y1
meets X;
defpred P4[set] means ex Y1,Y2,Y3,Y4 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4
in $1 & Y1 meets X;
defpred P3[set] means ex Y1,Y2,Y3,Y4,Y5 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 &
Y4 in Y5 & Y5 in $1 & Y1 meets X;
defpred P2[set] means ex Y1,Y2,Y3,Y4,Y5,Y6 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4
& Y4 in Y5 & Y5 in Y6 & Y6 in $1 & Y1 meets X;
consider Z2 such that
A2: for Y holds Y in Z2 iff Y in union union X & P2[Y] from XBOOLE_0:sch
1;
consider Z7 such that
A3: for Y holds Y in Z7 iff Y in union union union union union union
union X & P7[Y] from XBOOLE_0:sch 1;
consider Z6 such that
A4: for Y holds Y in Z6 iff Y in union union union union union union X &
P6[Y] from XBOOLE_0:sch 1;
consider Z8 such that
A5: for Y holds Y in Z8 iff Y in union union union union union union
union union X & P8[Y] from XBOOLE_0:sch 1;
consider Z3 such that
A6: for Y holds Y in Z3 iff Y in union union union X & P3[Y] from
XBOOLE_0:sch 1;
consider Z5 such that
A7: for Y holds Y in Z5 iff Y in union union union union union X & P5[Y]
from XBOOLE_0:sch 1;
consider Z4 such that
A8: for Y holds Y in Z4 iff Y in union union union union X & P4[Y] from
XBOOLE_0:sch 1;
set V = X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8;
assume X <> {};
then consider Y such that
A9: Y in V and
A10: Y misses V by MCART_1:1;
A11: V = X \/ (Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5) \/ Z6 \/ Z7 \/ Z8 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6) \/ Z7 \/ Z8 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7) \/ Z8 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8) by XBOOLE_1:4;
A12: now
assume
A13: Y in Z1;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7 such that
A14: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 and
A15: Y7 in Y and
A16: Y1 meets X by A1;
Y in union X by A1,A13;
then Y7 in union union X by A15,TARSKI:def 4;
then Y7 in Z2 by A2,A14,A16;
then Y7 in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
then Y meets X \/ Z1 \/ Z2 by A15,XBOOLE_0:3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:70;
hence contradiction by A10,XBOOLE_1:70;
end;
A17: now
assume
A18: Y in Z2;
then consider Y1,Y2,Y3,Y4,Y5,Y6 such that
A19: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 and
A20: Y6 in Y and
A21: Y1 meets X by A2;
Y in union union X by A2,A18;
then Y6 in union union union X by A20,TARSKI:def 4;
then Y6 in Z3 by A6,A19,A21;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 3;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y6 in V by XBOOLE_0:def 3;
hence contradiction by A10,A20,XBOOLE_0:3;
end;
A22: now
assume
A23: Y in Z3;
then consider Y1,Y2,Y3,Y4,Y5 such that
A24: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 and
A25: Y5 in Y and
A26: Y1 meets X by A6;
Y in union union union X by A6,A23;
then Y5 in union union union union X by A25,TARSKI:def 4;
then Y5 in Z4 by A8,A24,A26;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y5 in V by XBOOLE_0:def 3;
hence contradiction by A10,A25,XBOOLE_0:3;
end;
assume
A27: not thesis;
now
assume
A28: Y in X;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8 such that
A29: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 &
Y7 in Y8 and
A30: Y8 in Y and
A31: not Y1 misses X by A27;
Y8 in union X by A28,A30,TARSKI:def 4;
then Y8 in Z1 by A1,A29,A31;
then Y8 in X \/ Z1 by XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by A30,XBOOLE_0:3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:70;
hence contradiction by A10,XBOOLE_1:70;
end;
then Y in Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by A11,A9,
XBOOLE_0:def 3;
then Y in Z1 \/ (Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5) \/ Z6 \/ Z7 \/ Z8 by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6) \/ Z7 \/ Z8 by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7) \/ Z8 by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8) by XBOOLE_1:4;
then Y in Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by A12,XBOOLE_0:def 3;
then Y in Z2 \/ (Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 by XBOOLE_1:4;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5) \/ Z6 \/ Z7 \/ Z8 by XBOOLE_1:4;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5 \/ Z6) \/ Z7 \/ Z8 by XBOOLE_1:4;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7) \/ Z8 by XBOOLE_1:4;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8) by XBOOLE_1:4;
then Y in Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by A17,XBOOLE_0:def 3;
then Y in Z3 \/ (Z4 \/ Z5) \/ Z6 \/ Z7 \/ Z8 by XBOOLE_1:4;
then Y in Z3 \/ (Z4 \/ Z5 \/ Z6) \/ Z7 \/ Z8 by XBOOLE_1:4;
then Y in Z3 \/ (Z4 \/ Z5 \/ Z6 \/ Z7) \/ Z8 by XBOOLE_1:4;
then Y in Z3 \/ (Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8) by XBOOLE_1:4;
then Y in Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by A22,XBOOLE_0:def 3;
then Y in Z4 \/ (Z5 \/ Z6) \/ Z7 \/ Z8 by XBOOLE_1:4;
then Y in Z4 \/ (Z5 \/ Z6 \/ Z7) \/ Z8 by XBOOLE_1:4;
then
A32: Y in Z4 \/ (Z5 \/ Z6 \/ Z7 \/ Z8) by XBOOLE_1:4;
A33: now
assume
A34: Y in Z5;
then consider Y1,Y2,Y3 such that
A35: Y1 in Y2 & Y2 in Y3 and
A36: Y3 in Y and
A37: Y1 meets X by A7;
Y in union union union union union X by A7,A34;
then Y3 in union union union union union union X by A36,TARSKI:def 4;
then Y3 in Z6 by A4,A35,A37;
then Y3 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then Y3 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y3 in V by XBOOLE_0:def 3;
hence contradiction by A10,A36,XBOOLE_0:3;
end;
A38: now
assume
A39: Y in Z6;
then consider Y1,Y2 such that
A40: Y1 in Y2 and
A41: Y2 in Y and
A42: Y1 meets X by A4;
Y in union union union union union union X by A4,A39;
then Y2 in union union union union union union union X by A41,TARSKI:def 4;
then Y2 in Z7 by A3,A40,A42;
then Y2 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y2 in V by XBOOLE_0:def 3;
hence contradiction by A10,A41,XBOOLE_0:3;
end;
A43: now
assume
A44: Y in Z7;
then consider Y1 such that
A45: Y1 in Y and
A46: Y1 meets X by A3;
Y in union union union union union union union X by A3,A44;
then
Y1 in union union union union union union union union X by A45,TARSKI:def 4;
then Y1 in Z8 by A5,A46;
then Y1 in V by XBOOLE_0:def 3;
hence contradiction by A10,A45,XBOOLE_0:3;
end;
now
assume
A47: Y in Z4;
then consider Y1,Y2,Y3,Y4 such that
A48: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 and
A49: Y4 in Y and
A50: Y1 meets X by A8;
Y in union union union union X by A8,A47;
then Y4 in union union union union union X by A49,TARSKI:def 4;
then Y4 in Z5 by A7,A48,A50;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y4 in V by XBOOLE_0:def 3;
hence contradiction by A10,A49,XBOOLE_0:3;
end;
then Y in Z5 \/ Z6 \/ Z7 \/ Z8 by A32,XBOOLE_0:def 3;
then Y in Z5 \/ (Z6 \/ Z7) \/ Z8 by XBOOLE_1:4;
then Y in Z5 \/ (Z6 \/ Z7 \/ Z8) by XBOOLE_1:4;
then Y in Z6 \/ Z7 \/ Z8 by A33,XBOOLE_0:def 3;
then Y in Z6 \/ (Z7 \/ Z8) by XBOOLE_1:4;
then Y in Z7 \/ Z8 by A38,XBOOLE_0:def 3;
then Y in Z8 by A43,XBOOLE_0:def 3;
then Y meets X by A5;
hence contradiction by A11,A10,XBOOLE_1:70;
end;
theorem Th40:
X <> {} implies ex Y st Y in X & for Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9
st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 &
Y8 in Y9 & Y9 in Y holds Y1 misses X
proof
defpred P1[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8 st Y1 in Y2 & Y2 in Y3 & Y3
in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in $1 & Y1 meets X;
consider Z1 such that
A1: for Y holds Y in Z1 iff Y in union X & P1[Y] from XBOOLE_0:sch 1;
defpred P9[set] means $1 meets X;
defpred P8[set] means ex Y1 st Y1 in $1 & Y1 meets X;
defpred P7[set] means ex Y1,Y2 st Y1 in Y2 & Y2 in $1 & Y1 meets X;
defpred P6[set] means ex Y1,Y2,Y3 st Y1 in Y2 & Y2 in Y3 & Y3 in $1 & Y1
meets X;
defpred P5[set] means ex Y1,Y2,Y3,Y4 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4
in $1 & Y1 meets X;
defpred P4[set] means ex Y1,Y2,Y3,Y4,Y5 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 &
Y4 in Y5 & Y5 in $1 & Y1 meets X;
defpred P3[set] means ex Y1,Y2,Y3,Y4,Y5,Y6 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4
& Y4 in Y5 & Y5 in Y6 & Y6 in $1 & Y1 meets X;
defpred P2[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7 st Y1 in Y2 & Y2 in Y3 & Y3 in
Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in $1 & Y1 meets X;
consider Z2 such that
A2: for Y holds Y in Z2 iff Y in union union X & P2[Y] from XBOOLE_0:sch
1;
consider Z8 such that
A3: for Y holds Y in Z8 iff Y in union union union union union union
union union X & P8[Y] from XBOOLE_0:sch 1;
consider Z5 such that
A4: for Y holds Y in Z5 iff Y in union union union union union X & P5[Y]
from XBOOLE_0:sch 1;
consider Z4 such that
A5: for Y holds Y in Z4 iff Y in union union union union X & P4[Y] from
XBOOLE_0:sch 1;
consider Z9 such that
A6: for Y holds Y in Z9 iff Y in union union union union union union
union union union X & P9[Y] from XBOOLE_0:sch 1;
consider Z3 such that
A7: for Y holds Y in Z3 iff Y in union union union X & P3[Y] from
XBOOLE_0:sch 1;
consider Z7 such that
A8: for Y holds Y in Z7 iff Y in union union union union union union
union X & P7[Y] from XBOOLE_0:sch 1;
consider Z6 such that
A9: for Y holds Y in Z6 iff Y in union union union union union union X &
P6[Y] from XBOOLE_0:sch 1;
set V = X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9;
assume X <> {};
then consider Y such that
A10: Y in V and
A11: Y misses V by MCART_1:1;
A12: V = X \/ (Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5) \/ Z6 \/ Z7 \/ Z8 \/ Z9 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6) \/ Z7 \/ Z8 \/ Z9 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7) \/ Z8 \/ Z9 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8) \/ Z9 by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9) by XBOOLE_1:4;
A13: now
assume
A14: Y in Z1;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8 such that
A15: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 &
Y7 in Y8 and
A16: Y8 in Y and
A17: Y1 meets X by A1;
Y in union X by A1,A14;
then Y8 in union union X by A16,TARSKI:def 4;
then Y8 in Z2 by A2,A15,A17;
then Y8 in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
then Y meets X \/ Z1 \/ Z2 by A16,XBOOLE_0:3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_1:70;
hence contradiction by A11,XBOOLE_1:70;
end;
A18: now
assume
A19: Y in Z2;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7 such that
A20: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 and
A21: Y7 in Y and
A22: Y1 meets X by A2;
Y in union union X by A2,A19;
then Y7 in union union union X by A21,TARSKI:def 4;
then Y7 in Z3 by A7,A20,A22;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y7 in V by XBOOLE_0:def 3;
hence contradiction by A11,A21,XBOOLE_0:3;
end;
assume
A23: not thesis;
now
assume
A24: Y in X;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9 such that
A25: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 &
Y7 in Y8 & Y8 in Y9 and
A26: Y9 in Y and
A27: not Y1 misses X by A23;
Y9 in union X by A24,A26,TARSKI:def 4;
then Y9 in Z1 by A1,A25,A27;
then Y9 in X \/ Z1 by XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by A26,XBOOLE_0:3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_1:70;
hence contradiction by A11,XBOOLE_1:70;
end;
then Y in Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by A12,A10,
XBOOLE_0:def 3;
then Y in Z1 \/ (Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5) \/ Z6 \/ Z7 \/ Z8 \/ Z9 by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6) \/ Z7 \/ Z8 \/ Z9 by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7) \/ Z8 \/ Z9 by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8) \/ Z9 by XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9) by XBOOLE_1:4;
then Y in Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by A13,XBOOLE_0:def 3;
then Y in Z2 \/ (Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by XBOOLE_1:4;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5) \/ Z6 \/ Z7 \/ Z8 \/ Z9 by XBOOLE_1:4;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5 \/ Z6) \/ Z7 \/ Z8 \/ Z9 by XBOOLE_1:4;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7) \/ Z8 \/ Z9 by XBOOLE_1:4;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8) \/ Z9 by XBOOLE_1:4;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9) by XBOOLE_1:4;
then Y in Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by A18,XBOOLE_0:def 3;
then Y in Z3 \/ (Z4 \/ Z5) \/ Z6 \/ Z7 \/ Z8 \/ Z9 by XBOOLE_1:4;
then Y in Z3 \/ (Z4 \/ Z5 \/ Z6) \/ Z7 \/ Z8 \/ Z9 by XBOOLE_1:4;
then Y in Z3 \/ (Z4 \/ Z5 \/ Z6 \/ Z7) \/ Z8 \/ Z9 by XBOOLE_1:4;
then Y in Z3 \/ (Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8) \/ Z9 by XBOOLE_1:4;
then
A28: Y in Z3 \/ (Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9) by XBOOLE_1:4;
A29: now
assume
A30: Y in Z4;
then consider Y1,Y2,Y3,Y4,Y5 such that
A31: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 and
A32: Y5 in Y and
A33: Y1 meets X by A5;
Y in union union union union X by A5,A30;
then Y5 in union union union union union X by A32,TARSKI:def 4;
then Y5 in Z5 by A4,A31,A33;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y5 in V by XBOOLE_0:def 3;
hence contradiction by A11,A32,XBOOLE_0:3;
end;
A34: now
assume
A35: Y in Z5;
then consider Y1,Y2,Y3,Y4 such that
A36: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 and
A37: Y4 in Y and
A38: Y1 meets X by A4;
Y in union union union union union X by A4,A35;
then Y4 in union union union union union union X by A37,TARSKI:def 4;
then Y4 in Z6 by A9,A36,A38;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y4 in V by XBOOLE_0:def 3;
hence contradiction by A11,A37,XBOOLE_0:3;
end;
A39: now
assume
A40: Y in Z6;
then consider Y1,Y2,Y3 such that
A41: Y1 in Y2 & Y2 in Y3 and
A42: Y3 in Y and
A43: Y1 meets X by A9;
Y in union union union union union union X by A9,A40;
then Y3 in union union union union union union union X by A42,TARSKI:def 4;
then Y3 in Z7 by A8,A41,A43;
then Y3 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y3 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y3 in V by XBOOLE_0:def 3;
hence contradiction by A11,A42,XBOOLE_0:3;
end;
A44: now
assume
A45: Y in Z7;
then consider Y1,Y2 such that
A46: Y1 in Y2 and
A47: Y2 in Y and
A48: Y1 meets X by A8;
Y in union union union union union union union X by A8,A45;
then
Y2 in union union union union union union union union X by A47,TARSKI:def 4;
then Y2 in Z8 by A3,A46,A48;
then Y2 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y2 in V by XBOOLE_0:def 3;
hence contradiction by A11,A47,XBOOLE_0:3;
end;
A49: now
assume
A50: Y in Z8;
then consider Y1 such that
A51: Y1 in Y and
A52: Y1 meets X by A3;
Y in union union union union union union union union X by A3,A50;
then Y1 in union union union union union union union union union X by A51,
TARSKI:def 4;
then Y1 in Z9 by A6,A52;
then Y1 in V by XBOOLE_0:def 3;
hence contradiction by A11,A51,XBOOLE_0:3;
end;
now
assume
A53: Y in Z3;
then consider Y1,Y2,Y3,Y4,Y5,Y6 such that
A54: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 and
A55: Y6 in Y and
A56: Y1 meets X by A7;
Y in union union union X by A7,A53;
then Y6 in union union union union X by A55,TARSKI:def 4;
then Y6 in Z4 by A5,A54,A56;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y6 in V by XBOOLE_0:def 3;
hence contradiction by A11,A55,XBOOLE_0:3;
end;
then Y in Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by A28,XBOOLE_0:def 3;
then Y in Z4 \/ (Z5 \/ Z6) \/ Z7 \/ Z8 \/ Z9 by XBOOLE_1:4;
then Y in Z4 \/ (Z5 \/ Z6 \/ Z7) \/ Z8 \/ Z9 by XBOOLE_1:4;
then Y in Z4 \/ (Z5 \/ Z6 \/ Z7 \/ Z8) \/ Z9 by XBOOLE_1:4;
then Y in Z4 \/ (Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9) by XBOOLE_1:4;
then Y in Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by A29,XBOOLE_0:def 3;
then Y in Z5 \/ (Z6 \/ Z7) \/ Z8 \/ Z9 by XBOOLE_1:4;
then Y in Z5 \/ (Z6 \/ Z7 \/ Z8) \/ Z9 by XBOOLE_1:4;
then Y in Z5 \/ (Z6 \/ Z7 \/ Z8 \/ Z9) by XBOOLE_1:4;
then Y in Z6 \/ Z7 \/ Z8 \/ Z9 by A34,XBOOLE_0:def 3;
then Y in Z6 \/ (Z7 \/ Z8) \/ Z9 by XBOOLE_1:4;
then Y in Z6 \/ (Z7 \/ Z8 \/ Z9) by XBOOLE_1:4;
then Y in Z7 \/ Z8 \/ Z9 by A39,XBOOLE_0:def 3;
then Y in Z7 \/ (Z8 \/ Z9) by XBOOLE_1:4;
then Y in Z8 \/ Z9 by A44,XBOOLE_0:def 3;
then Y in Z9 by A49,XBOOLE_0:def 3;
then Y meets X by A6;
hence contradiction by A12,A11,XBOOLE_1:70;
end;
::
:: Tuples for n=6
::
definition
let x1,x2,x3,x4,x5,x6;
func [x1,x2,x3,x4,x5,x6] equals
[[x1,x2,x3,x4,x5],x6];
correctness;
end;
theorem Th41:
[x1,x2,x3,x4,x5,x6] = [[[[[x1,x2],x3],x4],x5],x6]
proof
thus [x1,x2,x3,x4,x5,x6] = [[[[x1,x2,x3],x4],x5],x6] by MCART_1:def 4
.= [[[[[x1,x2],x3],x4],x5],x6] by MCART_1:def 3;
end;
theorem
[x1,x2,x3,x4,x5,x6] = [[x1,x2,x3,x4],x5,x6] by MCART_1:def 3;
theorem
[x1,x2,x3,x4,x5,x6] = [[x1,x2,x3],x4,x5,x6]
proof
thus [x1,x2,x3,x4,x5,x6] = [[[[[x1,x2],x3],x4],x5],x6] by Th41
.= [[[x1,x2],x3],x4,x5,x6] by MCART_1:27
.= [[x1,x2,x3],x4,x5,x6] by MCART_1:def 3;
end;
theorem
[x1,x2,x3,x4,x5,x6] = [[x1,x2],x3,x4,x5,x6] by Th6;
theorem Th45:
[x1,x2,x3,x4,x5,x6] = [y1,y2,y3,y4,y5,y6] implies x1 = y1 & x2 =
y2 & x3 = y3 & x4 = y4 & x5 = y5 & x6 = y6
proof
assume
A1: [x1,x2,x3,x4,x5,x6] = [y1,y2,y3,y4,y5,y6];
then [x1,x2,x3,x4,x5] = [y1,y2,y3,y4,y5] by ZFMISC_1:27;
hence thesis by A1,Th7,ZFMISC_1:27;
end;
theorem Th46:
X <> {} implies ex v st v in X & not ex x1,x2,x3,x4,x5,x6 st (x1
in X or x2 in X) & v = [x1,x2,x3,x4,x5,x6]
proof
assume X <> {};
then consider Y such that
A1: Y in X and
A2: for Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 &
Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in Y holds Y1 misses
X by Th40;
take v = Y;
thus v in X by A1;
given x1,x2,x3,x4,x5,x6 such that
A3: x1 in X or x2 in X and
A4: v = [x1,x2,x3,x4,x5,x6];
set Y1 = { x1, x2 }, Y2 = { Y1, {x1} }, Y3 = { Y2, x3 }, Y4 = { Y3, {Y2} },
Y5 = { Y4, x4 }, Y6 = { Y5, {Y4} }, Y7 = { Y6, x5 }, Y8 = { Y7, {Y6} }, Y9 = {
Y8, x6 };
A5: Y3 in Y4 & Y4 in Y5 by TARSKI:def 2;
x1 in Y1 & x2 in Y1 by TARSKI:def 2;
then
A6: not Y1 misses X by A3,XBOOLE_0:3;
A7: Y7 in Y8 & Y8 in Y9 by TARSKI:def 2;
Y = [[[[[x1,x2],x3],x4],x5],x6] by A4,Th41
.= [[[[ Y2,x3],x4],x5],x6 ] by TARSKI:def 5
.= [[[ Y4,x4],x5],x6 ] by TARSKI:def 5
.= [[ Y6,x5 ],x6] by TARSKI:def 5
.= [ Y8,x6 ] by TARSKI:def 5
.= { Y9, { Y8 } } by TARSKI:def 5;
then
A8: Y9 in Y by TARSKI:def 2;
A9: Y5 in Y6 & Y6 in Y7 by TARSKI:def 2;
Y1 in Y2 & Y2 in Y3 by TARSKI:def 2;
hence contradiction by A2,A6,A5,A9,A7,A8;
end;
::
:: Cartesian products of six sets
::
definition
let X1,X2,X3,X4,X5,X6;
func [:X1,X2,X3,X4,X5,X6:] -> set equals
[:[: X1,X2,X3,X4,X5 :],X6 :];
coherence;
end;
theorem Th47:
[:X1,X2,X3,X4,X5,X6:] = [:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:]
proof
thus [:X1,X2,X3,X4,X5,X6:] = [:[:[:[:X1,X2,X3:],X4:],X5:],X6:] by
ZFMISC_1:def 4
.= [:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:] by ZFMISC_1:def 3;
end;
theorem
[:X1,X2,X3,X4,X5,X6:] = [:[:X1,X2,X3,X4:],X5,X6:] by ZFMISC_1:def 3;
theorem
[:X1,X2,X3,X4,X5,X6:] = [:[:X1,X2,X3:],X4,X5,X6:]
proof
thus [:X1,X2,X3,X4,X5,X6:] = [:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:] by Th47
.= [:[:[:X1,X2:],X3:],X4,X5,X6:] by MCART_1:49
.= [:[:X1,X2,X3:],X4,X5,X6:] by ZFMISC_1:def 3;
end;
theorem
[:X1,X2,X3,X4,X5,X6:] = [:[:X1,X2:],X3,X4,X5,X6:] by Th12;
theorem Th51:
X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {}
iff [:X1,X2,X3,X4,X5,X6:] <> {}
proof
X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} iff [:X1,X2,X3,X4,
X5:] <> {} by Th13;
hence thesis by ZFMISC_1:90;
end;
theorem Th52:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} implies ( [:
X1,X2,X3,X4,X5,X6:] = [:Y1,Y2,Y3,Y4,Y5,Y6:] implies X1=Y1 & X2=Y2 & X3=Y3 & X4=
Y4 & X5=Y5 & X6=Y6 )
proof
assume
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{};
then
A2: [:X1,X2,X3,X4,X5:] <> {} by Th13;
assume
A3: X6<>{};
assume
A4: [:X1,X2,X3,X4,X5,X6:] = [:Y1,Y2,Y3,Y4,Y5,Y6:];
then [:X1,X2,X3,X4,X5:] = [:Y1,Y2,Y3,Y4,Y5:] by A2,A3,ZFMISC_1:110;
hence thesis by A1,A2,A3,A4,Th14,ZFMISC_1:110;
end;
theorem
[:X1,X2,X3,X4,X5,X6:]<>{} & [:X1,X2,X3,X4,X5,X6:] = [:Y1,Y2,Y3,Y4,Y5,
Y6:] implies X1=Y1 & X2=Y2 & X3=Y3 & X4=Y4 & X5=Y5 & X6=Y6
proof
assume
A1: [:X1,X2,X3,X4,X5,X6:]<>{};
then
A2: X3<>{} & X4<>{} by Th51;
A3: X5<>{} & X6<>{} by A1,Th51;
X1<>{} & X2<>{} by A1,Th51;
hence thesis by A2,A3,Th52;
end;
theorem
[:X,X,X,X,X,X:] = [:Y,Y,Y,Y,Y,Y:] implies X = Y
proof
assume [:X,X,X,X,X,X:] = [:Y,Y,Y,Y,Y,Y:];
then X<>{} or Y<>{} implies thesis by Th52;
hence thesis;
end;
theorem Th55:
X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {}
implies for x being Element of [:X1,X2,X3,X4,X5,X6:] ex xx1,xx2,xx3,xx4,xx5,xx6
st x = [xx1,xx2,xx3,xx4,xx5,xx6]
proof
assume that
A1: X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} and
A2: X6 <> {};
let x be Element of [:X1,X2,X3,X4,X5,X6:];
reconsider x9=x as Element of [:[:X1,X2,X3,X4,X5:],X6:];
[:X1,X2,X3,X4,X5:] <> {} by A1,Th13;
then consider x12345 being (Element of [:X1,X2,X3,X4,X5:]), xx6 such that
A3: x9 = [x12345,xx6] by A2,Lm1;
consider xx1,xx2,xx3,xx4,xx5 such that
A4: x12345 = [xx1,xx2,xx3,xx4,xx5] by A1,Th17;
take xx1,xx2,xx3,xx4,xx5,xx6;
thus thesis by A3,A4;
end;
definition
let X1,X2,X3,X4,X5,X6;
assume
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{};
let x be Element of [:X1,X2,X3,X4,X5,X6:];
func x`1 -> Element of X1 means
:Def10:
x = [x1,x2,x3,x4,x5,x6] implies it = x1;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6 such that
A2: x = [xx1,xx2,xx3,xx4,xx5,xx6] by A1,Th55;
take xx1;
thus thesis by A2,Th45;
end;
uniqueness
proof
let y,z be Element of X1;
assume
A3: x = [x1,x2,x3,x4,x5,x6] implies y = x1;
assume
A4: x = [x1,x2,x3,x4,x5,x6] implies z = x1;
consider xx1,xx2,xx3,xx4,xx5,xx6 such that
A5: x = [xx1,xx2,xx3,xx4,xx5,xx6] by A1,Th55;
y = xx1 by A5,A3;
hence thesis by A5,A4;
end;
func x`2 -> Element of X2 means
:Def11:
x = [x1,x2,x3,x4,x5,x6] implies it = x2;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6 such that
A6: x = [xx1,xx2,xx3,xx4,xx5,xx6] by A1,Th55;
take xx2;
thus thesis by A6,Th45;
end;
uniqueness
proof
let y,z be Element of X2;
assume
A7: x = [x1,x2,x3,x4,x5,x6] implies y = x2;
assume
A8: x = [x1,x2,x3,x4,x5,x6] implies z = x2;
consider xx1,xx2,xx3,xx4,xx5,xx6 such that
A9: x = [xx1,xx2,xx3,xx4,xx5,xx6] by A1,Th55;
y = xx2 by A9,A7;
hence thesis by A9,A8;
end;
func x`3 -> Element of X3 means
:Def12:
x = [x1,x2,x3,x4,x5,x6] implies it = x3;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6 such that
A10: x = [xx1,xx2,xx3,xx4,xx5,xx6] by A1,Th55;
take xx3;
thus thesis by A10,Th45;
end;
uniqueness
proof
let y,z be Element of X3;
assume
A11: x = [x1,x2,x3,x4,x5,x6] implies y = x3;
assume
A12: x = [x1,x2,x3,x4,x5,x6] implies z = x3;
consider xx1,xx2,xx3,xx4,xx5,xx6 such that
A13: x = [xx1,xx2,xx3,xx4,xx5,xx6] by A1,Th55;
y = xx3 by A13,A11;
hence thesis by A13,A12;
end;
func x`4 -> Element of X4 means
:Def13:
x = [x1,x2,x3,x4,x5,x6] implies it = x4;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6 such that
A14: x = [xx1,xx2,xx3,xx4,xx5,xx6] by A1,Th55;
take xx4;
thus thesis by A14,Th45;
end;
uniqueness
proof
let y,z be Element of X4;
assume
A15: x = [x1,x2,x3,x4,x5,x6] implies y = x4;
assume
A16: x = [x1,x2,x3,x4,x5,x6] implies z = x4;
consider xx1,xx2,xx3,xx4,xx5,xx6 such that
A17: x = [xx1,xx2,xx3,xx4,xx5,xx6] by A1,Th55;
y = xx4 by A17,A15;
hence thesis by A17,A16;
end;
func x`5 -> Element of X5 means
:Def14:
x = [x1,x2,x3,x4,x5,x6] implies it = x5;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6 such that
A18: x = [xx1,xx2,xx3,xx4,xx5,xx6] by A1,Th55;
take xx5;
thus thesis by A18,Th45;
end;
uniqueness
proof
let y,z be Element of X5;
assume
A19: x = [x1,x2,x3,x4,x5,x6] implies y = x5;
assume
A20: x = [x1,x2,x3,x4,x5,x6] implies z = x5;
consider xx1,xx2,xx3,xx4,xx5,xx6 such that
A21: x = [xx1,xx2,xx3,xx4,xx5,xx6] by A1,Th55;
y = xx5 by A21,A19;
hence thesis by A21,A20;
end;
func x`6 -> Element of X6 means
:Def15:
x = [x1,x2,x3,x4,x5,x6] implies it = x6;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6 such that
A22: x = [xx1,xx2,xx3,xx4,xx5,xx6] by A1,Th55;
take xx6;
thus thesis by A22,Th45;
end;
uniqueness
proof
let y,z be Element of X6;
assume
A23: x = [x1,x2,x3,x4,x5,x6] implies y = x6;
assume
A24: x = [x1,x2,x3,x4,x5,x6] implies z = x6;
consider xx1,xx2,xx3,xx4,xx5,xx6 such that
A25: x = [xx1,xx2,xx3,xx4,xx5,xx6] by A1,Th55;
y = xx6 by A25,A23;
hence thesis by A25,A24;
end;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} implies for x
being Element of [:X1,X2,X3,X4,X5,X6:] for x1,x2,x3,x4,x5,x6 st x = [x1,x2,x3,
x4,x5,x6] holds x`1 = x1 & x`2 = x2 & x`3 = x3 & x`4 = x4 & x`5 = x5 & x`6 = x6
by Def10,Def11,Def12,Def13,Def14,Def15;
theorem Th57:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} implies for
x being Element of [:X1,X2,X3,X4,X5,X6:] holds x = [x`1,x`2,x`3,x`4,x`5,x`6]
proof
assume
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{};
let x be Element of [:X1,X2,X3,X4,X5,X6:];
consider xx1,xx2,xx3,xx4,xx5,xx6 such that
A2: x = [xx1,xx2,xx3,xx4,xx5,xx6] by A1,Th55;
thus x = [x`1,xx2,xx3,xx4,xx5,xx6] by A1,A2,Def10
.= [x`1,x`2,xx3,xx4,xx5,xx6] by A1,A2,Def11
.= [x`1,x`2,x`3,xx4,xx5,xx6] by A1,A2,Def12
.= [x`1,x`2,x`3,x`4,xx5,xx6] by A1,A2,Def13
.= [x`1,x`2,x`3,x`4,x`5,xx6] by A1,A2,Def14
.= [x`1,x`2,x`3,x`4,x`5,x`6] by A1,A2,Def15;
end;
theorem Th58:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} implies for
x being Element of [:X1,X2,X3,X4,X5,X6:] holds x`1 = (x qua set)`1`1`1`1`1 & x
`2 = (x qua set)`1`1`1`1`2 & x`3 = (x qua set)`1`1`1`2 & x`4 = (x qua set)`1`1
`2 & x`5 = (x qua set)`1`2 & x`6 = (x qua set)`2
proof
assume
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{};
let x be Element of [:X1,X2,X3,X4,X5,X6:];
thus x`1 = [ x`1, x`2]`1 by MCART_1:7
.= [[x`1, x`2],x`3]`1`1 by MCART_1:7
.= [ x`1, x`2 ,x`3]`1`1 by MCART_1:def 3
.= [[x`1, x`2 ,x`3],x`4]`1`1`1 by MCART_1:7
.= [ x`1, x`2 ,x`3 ,x`4]`1`1`1 by MCART_1:def 4
.= [[x`1, x`2 ,x`3 ,x`4], x`5]`1`1`1`1 by MCART_1:7
.= [ x`1, x`2 ,x`3 ,x`4 ,x`5, x`6]`1`1`1`1`1 by MCART_1:7
.= (x qua set)`1`1`1`1`1 by A1,Th57;
thus x`2 = [ x`1, x`2]`2 by MCART_1:7
.= [[x`1, x`2],x`3]`1`2 by MCART_1:7
.= [ x`1, x`2, x`3]`1`2 by MCART_1:def 3
.= [[x`1, x`2, x`3],x`4]`1`1`2 by MCART_1:7
.= [ x`1, x`2, x`3, x`4]`1`1`2 by MCART_1:def 4
.= [[x`1, x`2, x`3, x`4 ], x`5]`1`1`1`2 by MCART_1:7
.= [ x`1, x`2, x`3, x`4, x`5, x`6]`1`1`1`1`2 by MCART_1:7
.= (x qua set)`1`1`1`1`2 by A1,Th57;
thus x`3 = [[x`1, x`2],x`3]`2 by MCART_1:7
.= [ x`1, x`2, x`3]`2 by MCART_1:def 3
.= [[x`1, x`2, x`3],x`4]`1`2 by MCART_1:7
.= [ x`1, x`2, x`3, x`4]`1`2 by MCART_1:def 4
.= [[x`1, x`2, x`3, x`4],x`5]`1`1`2 by MCART_1:7
.= [ x`1, x`2, x`3, x`4, x`5, x`6]`1`1`1`2 by MCART_1:7
.= (x qua set)`1`1`1`2 by A1,Th57;
thus x`4 = [[x`1,x`2,x`3],x`4]`2 by MCART_1:7
.= [ x`1,x`2,x`3, x`4]`2 by MCART_1:def 4
.= [[x`1,x`2,x`3, x`4],x`5]`1`2 by MCART_1:7
.= [ x`1,x`2,x`3, x`4, x`5, x`6]`1`1`2 by MCART_1:7
.= (x qua set)`1`1`2 by A1,Th57;
thus x`5 = [[x`1,x`2,x`3,x`4],x`5]`2 by MCART_1:7
.= [ x`1,x`2,x`3,x`4,x`5, x`6]`1`2 by MCART_1:7
.= (x qua set)`1`2 by A1,Th57;
thus x`6 = [ x`1,x`2,x`3,x`4,x`5, x`6]`2 by MCART_1:7
.= (x qua set)`2 by A1,Th57;
end;
theorem Th59:
X1 c= [:X1,X2,X3,X4,X5,X6:] or X1 c= [:X2,X3,X4,X5,X6,X1:] or X1
c= [:X3,X4,X5,X6,X1,X2:] or X1 c= [:X4,X5,X6,X1,X2,X3:] or X1 c= [:X5,X6,X1,X2,
X3,X4:] or X1 c= [:X6,X1,X2,X3,X4,X5:] implies X1 = {}
proof
assume that
A1: X1 c= [:X1,X2,X3,X4,X5,X6:] or X1 c= [:X2,X3,X4,X5,X6,X1:] or X1 c=
[:X3,X4,X5,X6,X1,X2:] or X1 c= [:X4,X5,X6,X1,X2,X3:] or X1 c= [:X5,X6,X1,X2,X3,
X4:] or X1 c= [:X6,X1,X2,X3,X4,X5:] and
A2: X1 <> {};
A3: [:X1,X2,X3,X4,X5,X6:]<>{} or [:X2,X3,X4,X5,X6,X1:]<>{} or [:X3,X4,X5,X6,
X1,X2:]<>{} or [:X4,X5,X6,X1,X2,X3:]<>{} or [:X5,X6,X1,X2,X3,X4:]<>{} or [:X6,
X1,X2,X3,X4,X5:]<>{} by A1,A2,XBOOLE_1:3;
then
A4: X4<>{} & X5<>{} by Th51;
A5: X6<>{} by A3,Th51;
A6: X2<>{} & X3<>{} by A3,Th51;
now
per cases by A1;
suppose
A7: X1 c= [:X1,X2,X3,X4,X5,X6:];
consider v such that
A8: v in X1 and
A9: for x1,x2,x3,x4,x5,x6 st x1 in X1 or x2 in X1 holds v <> [x1,x2
,x3,x4,x5,x6] by A2,Th46;
reconsider v as Element of [:X1,X2,X3,X4,X5,X6:] by A7,A8;
v = [v`1,v`2,v`3,v`4,v`5,v`6] by A2,A6,A4,A5,Th57;
hence contradiction by A2,A9;
end;
suppose
X1 c= [:X2,X3,X4,X5,X6,X1:];
then X1 c= [:[:X2,X3:],X4,X5,X6,X1:] by Th12;
hence thesis by A2,Th21;
end;
suppose
X1 c= [:X3,X4,X5,X6,X1,X2:];
then X1 c= [:[:X3,X4:],X5,X6,X1,X2:] by Th12;
hence thesis by A2,Th21;
end;
suppose
X1 c= [:X4,X5,X6,X1,X2,X3:];
then X1 c= [:[:X4,X5:],X6,X1,X2,X3:] by Th12;
hence thesis by A2,Th21;
end;
suppose
X1 c= [:X5,X6,X1,X2,X3,X4:];
then X1 c= [:[:X5,X6:],X1,X2,X3,X4:] by Th12;
hence thesis by A2,Th21;
end;
suppose
A10: X1 c= [:X6,X1,X2,X3,X4,X5:];
consider v such that
A11: v in X1 and
A12: for x1,x2,x3,x4,x5,x6 st x1 in X1 or x2 in X1 holds v <> [x1,x2
,x3,x4,x5,x6] by A2,Th46;
reconsider v as Element of [:X6,X1,X2,X3,X4,X5:] by A10,A11;
v = [v`1,v`2,v`3,v`4,v`5,v`6] by A2,A6,A4,A5,Th57;
hence thesis by A2,A12;
end;
end;
hence contradiction;
end;
theorem
[:X1,X2,X3,X4,X5,X6:] meets [:Y1,Y2,Y3,Y4,Y5,Y6:] implies X1 meets Y1
& X2 meets Y2 & X3 meets Y3 & X4 meets Y4 & X5 meets Y5 & X6 meets Y6
proof
assume
A1: [:X1,X2,X3,X4,X5,X6:] meets [:Y1,Y2,Y3,Y4,Y5,Y6:];
[:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:] = [:X1,X2,X3,X4,X5,X6:] & [:[:[:[:
[:Y1,Y2:],Y3:],Y4:],Y5:],Y6:] = [:Y1,Y2,Y3,Y4,Y5,Y6:] by Th47;
then
A2: [:[:[:[:X1,X2:],X3:],X4:],X5:] meets [:[:[:[:Y1,Y2:],Y3:],Y4:],Y5:] by A1,
ZFMISC_1:104;
then
A3: [:[:[:X1,X2:],X3:],X4:] meets [:[:[:Y1,Y2:],Y3:],Y4:] by ZFMISC_1:104;
then
A4: [:[:X1,X2:],X3:] meets [:[:Y1,Y2:],Y3:] by ZFMISC_1:104;
then [:X1,X2:] meets [:Y1,Y2:] by ZFMISC_1:104;
hence thesis by A1,A2,A3,A4,ZFMISC_1:104;
end;
theorem Th61:
[:{x1},{x2},{x3},{x4},{x5},{x6}:] = { [x1,x2,x3,x4,x5,x6] }
proof
thus [:{x1},{x2},{x3},{x4},{x5},{x6}:] = [:[:{x1},{x2}:],{x3},{x4},{x5},{x6}
:] by Th12
.= [:{[x1,x2]}, {x3},{x4},{x5},{x6}:] by ZFMISC_1:29
.= { [[x1,x2], x3, x4, x5, x6]} by Th23
.= { [x1,x2,x3,x4,x5,x6] } by Th6;
end;
:: 6 - Tuples
reserve x for Element of [:X1,X2,X3,X4,X5,X6:];
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} implies for x1,x2,
x3,x4,x5,x6 st x = [x1,x2,x3,x4,x5,x6] holds x`1 = x1 & x`2 = x2 & x`3 = x3 & x
`4 = x4 & x`5 = x5 & x`6 = x6 by Def10,Def11,Def12,Def13,Def14,Def15;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & (for xx1,xx2,xx3
,xx4,xx5,xx6 st x = [xx1,xx2,xx3,xx4,xx5,xx6] holds y1 = xx1) implies y1 =x`1
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6 st x = [xx1,xx2,xx3,xx4,xx5,xx6] holds
y1 = xx1;
x = [x`1,x`2,x`3,x`4,x`5,x`6] by A1,Th57;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & (for xx1,xx2,xx3
,xx4,xx5,xx6 st x = [xx1,xx2,xx3,xx4,xx5,xx6] holds y2 = xx2) implies y2 =x`2
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6 st x = [xx1,xx2,xx3,xx4,xx5,xx6] holds
y2 = xx2;
x = [x`1,x`2,x`3,x`4,x`5,x`6] by A1,Th57;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & (for xx1,xx2,xx3
,xx4,xx5,xx6 st x = [xx1,xx2,xx3,xx4,xx5,xx6] holds y3 = xx3) implies y3 =x`3
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6 st x = [xx1,xx2,xx3,xx4,xx5,xx6] holds
y3 = xx3;
x = [x`1,x`2,x`3,x`4,x`5,x`6] by A1,Th57;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & (for xx1,xx2,xx3
,xx4,xx5,xx6 st x = [xx1,xx2,xx3,xx4,xx5,xx6] holds y4 = xx4) implies y4 =x`4
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6 st x = [xx1,xx2,xx3,xx4,xx5,xx6] holds
y4 = xx4;
x = [x`1,x`2,x`3,x`4,x`5,x`6] by A1,Th57;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & (for xx1,xx2,xx3
,xx4,xx5,xx6 st x = [xx1,xx2,xx3,xx4,xx5,xx6] holds y5 = xx5) implies y5 =x`5
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6 st x = [xx1,xx2,xx3,xx4,xx5,xx6] holds
y5 = xx5;
x = [x`1,x`2,x`3,x`4,x`5,x`6] by A1,Th57;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & (for xx1,xx2,xx3
,xx4,xx5,xx6 st x = [xx1,xx2,xx3,xx4,xx5,xx6] holds y6 = xx6) implies y6 =x`6
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6 st x = [xx1,xx2,xx3,xx4,xx5,xx6] holds
y6 = xx6;
x = [x`1,x`2,x`3,x`4,x`5,x`6] by A1,Th57;
hence thesis by A2;
end;
theorem Th69:
z in [: X1,X2,X3,X4,X5,X6 :] implies ex x1,x2,x3,x4,x5,x6 st x1
in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & z = [x1,x2,x3,x4
,x5,x6]
proof
assume z in [: X1,X2,X3,X4,X5,X6 :];
then consider x12345, x6 being set such that
A1: x12345 in [:X1,X2,X3,X4,X5:] and
A2: x6 in X6 and
A3: z = [x12345,x6] by ZFMISC_1:def 2;
consider x1, x2, x3, x4, x5 such that
A4: x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 and
A5: x12345 = [x1,x2,x3,x4,x5] by A1,Th30;
z = [x1,x2,x3,x4,x5,x6] by A3,A5;
hence thesis by A2,A4;
end;
theorem Th70:
[x1,x2,x3,x4,x5,x6] in [: X1,X2,X3,X4,X5,X6 :] iff x1 in X1 & x2
in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6
proof
A1: [x1,x2] in [:X1,X2:] iff x1 in X1 & x2 in X2 by ZFMISC_1:87;
[:X1,X2,X3,X4,X5,X6:] = [:[:X1,X2:],X3,X4,X5,X6:] & [x1,x2,x3,x4,x5,x6]
= [[ x1,x2],x3,x4,x5,x6] by Th6,Th12;
hence thesis by A1,Th31;
end;
theorem
(for z holds z in Z iff ex x1,x2,x3,x4,x5,x6 st x1 in X1 & x2 in X2 &
x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & z = [x1,x2,x3,x4,x5,x6]) implies Z
= [: X1,X2,X3,X4,X5,X6 :]
proof
assume
A1: for z holds z in Z iff ex x1,x2,x3,x4,x5,x6 st x1 in X1 & x2 in X2 &
x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & z = [x1,x2,x3,x4,x5,x6];
now
let z;
thus z in Z implies z in [:[:X1,X2,X3,X4,X5:],X6:]
proof
assume z in Z;
then consider x1,x2,x3,x4,x5,x6 such that
A2: x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 and
A3: x6 in X6 & z = [x1,x2,x3,x4,x5,x6] by A1;
[x1,x2,x3,x4,x5] in [:X1,X2,X3,X4,X5:] by A2,Th31;
hence thesis by A3,ZFMISC_1:def 2;
end;
assume z in [:[:X1,X2,X3,X4,X5:],X6:];
then consider x12345,x6 being set such that
A4: x12345 in [:X1,X2,X3,X4,X5:] and
A5: x6 in X6 and
A6: z = [x12345,x6] by ZFMISC_1:def 2;
consider x1,x2,x3,x4,x5 such that
A7: x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 and
A8: x12345 = [x1,x2,x3,x4,x5] by A4,Th30;
z = [x1,x2,x3,x4,x5,x6] by A6,A8;
hence z in Z by A1,A5,A7;
end;
hence thesis by TARSKI:1;
end;
theorem Th72:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & Y1<>{} &
Y2<>{} & Y3<>{} & Y4<>{} & Y5<>{} & Y6<>{} implies for x being (Element of [:X1
,X2,X3,X4,X5,X6:]), y being Element of [:Y1,Y2,Y3,Y4,Y5,Y6:] holds x = y
implies x`1 = y`1 & x`2 = y`2 & x`3 = y`3 & x`4 = y`4 & x`5 = y`5 & x`6 = y`6
proof
assume that
A1: X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} and
A2: Y1 <> {} & Y2 <> {} & Y3 <> {} & Y4 <> {} & Y5 <> {} & Y6 <> {};
let x be Element of [:X1,X2,X3,X4,X5,X6:];
let y be Element of [:Y1,Y2,Y3,Y4,Y5,Y6:];
assume
A3: x = y;
thus x`1 = (x qua set)`1`1`1`1`1 by A1,Th58
.= y`1 by A2,A3,Th58;
thus x`2 = (x qua set)`1`1`1`1`2 by A1,Th58
.= y`2 by A2,A3,Th58;
thus x`3 = (x qua set)`1`1`1`2 by A1,Th58
.= y`3 by A2,A3,Th58;
thus x`4 = (x qua set)`1`1`2 by A1,Th58
.= y`4 by A2,A3,Th58;
thus x`5 = (x qua set)`1`2 by A1,Th58
.= y`5 by A2,A3,Th58;
thus x`6 = (x qua set)`2 by A1,Th58
.= y`6 by A2,A3,Th58;
end;
theorem
for x being Element of [:X1,X2,X3,X4,X5,X6:] st x in [:A1,A2,A3,A4,A5,
A6:] holds x`1 in A1 & x`2 in A2 & x`3 in A3 & x`4 in A4 & x`5 in A5 & x`6 in
A6
proof
let x be Element of [:X1,X2,X3,X4,X5,X6:];
assume
A1: x in [:A1,A2,A3,A4,A5,A6:];
then reconsider y = x as Element of [:A1,A2,A3,A4,A5,A6:];
A2 <> {} by A1,Th51;
then
A2: y`2 in A2;
A6 <> {} by A1,Th51;
then
A3: y`6 in A6;
A5 <> {} by A1,Th51;
then
A4: y`5 in A5;
A4 <> {} by A1,Th51;
then
A5: y`4 in A4;
A3 <> {} by A1,Th51;
then
A6: y`3 in A3;
A1 <> {} by A1,Th51;
then y`1 in A1;
hence thesis by A2,A6,A5,A4,A3,Th72;
end;
theorem Th74:
X1 c= Y1 & X2 c= Y2 & X3 c= Y3 & X4 c= Y4 & X5 c= Y5 & X6 c= Y6
implies [:X1,X2,X3,X4,X5,X6:] c= [:Y1,Y2,Y3,Y4,Y5,Y6:]
proof
assume X1 c= Y1 & X2 c= Y2 & X3 c= Y3 & X4 c= Y4 & X5 c= Y5;
then
A1: [:X1,X2,X3,X4,X5:] c= [:Y1,Y2,Y3,Y4,Y5:] by Th35;
assume X6 c= Y6;
hence thesis by A1,ZFMISC_1:96;
end;
theorem
[:A1,A2,A3,A4,A5,A6:] is Subset of [:X1,X2,X3,X4,X5,X6:] by Th74;
begin :: Original MCART_4
theorem
X <> {} implies ex Y st Y in X & for Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA st
Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8
in Y9 & Y9 in YA & YA in Y holds Y1 misses X
proof
defpred P1[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9 st Y1 in Y2 & Y2 in Y3 &
Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in $1 & Y1
meets X;
consider Z1 such that
A1: for Y holds Y in Z1 iff Y in union X & P1[Y] from XBOOLE_0:sch 1;
defpred P9[set] means $1 meets X;
defpred P8[set] means ex Y1 st Y1 in $1 & Y1 meets X;
defpred P7[set] means ex Y1,Y2 st Y1 in Y2 & Y2 in $1 & Y1 meets X;
defpred P6[set] means ex Y1,Y2,Y3 st Y1 in Y2 & Y2 in Y3 & Y3 in $1 & Y1
meets X;
defpred P5[set] means ex Y1,Y2,Y3,Y4,Y5 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 &
Y4 in Y5 & Y5 in $1 & Y1 meets X;
defpred P4[set] means ex Y1,Y2,Y3,Y4,Y5,Y6 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4
& Y4 in Y5 & Y5 in Y6 & Y6 in $1 & Y1 meets X;
defpred P3[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7 st Y1 in Y2 & Y2 in Y3 & Y3 in
Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in $1 & Y1 meets X;
defpred P2[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8 st Y1 in Y2 & Y2 in Y3 & Y3
in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in $1 & Y1 meets X;
consider Z2 such that
A2: for Y holds Y in Z2 iff Y in union union X & P2[Y] from XBOOLE_0:sch
1;
consider Z5 such that
A3: for Y holds Y in Z5 iff Y in union union union union union X & P5[Y]
from XBOOLE_0:sch 1;
defpred P5[set] means ex Y1,Y2,Y3,Y4 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4
in $1 & Y1 meets X;
consider Z6 such that
A4: for Y holds Y in Z6 iff Y in union union union union union union X &
P5[Y] from XBOOLE_0:sch 1;
consider ZA such that
A5: for Y holds Y in ZA iff Y in union union union union union union
union union union union X & P9[Y] from XBOOLE_0:sch 1;
consider Z3 such that
A6: for Y holds Y in Z3 iff Y in union union union X & P3[Y] from
XBOOLE_0:sch 1;
consider Z7 such that
A7: for Y holds Y in Z7 iff Y in union union union union union union
union X & P6[Y] from XBOOLE_0:sch 1;
consider Z4 such that
A8: for Y holds Y in Z4 iff Y in union union union union X & P4[Y] from
XBOOLE_0:sch 1;
consider Z9 such that
A9: for Y holds Y in Z9 iff Y in union union union union union union
union union union X & P8[Y] from XBOOLE_0:sch 1;
consider Z8 such that
A10: for Y holds Y in Z8 iff Y in union union union union union union
union union X & P7[Y] from XBOOLE_0:sch 1;
set V = X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA;
assume X <> {};
then consider Y such that
A11: Y in V and
A12: Y misses V by MCART_1:1;
A13: V = X \/ (Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA by
XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA by
XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA by
XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5) \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA by
XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6) \/ Z7 \/ Z8 \/ Z9 \/ ZA by
XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7) \/ Z8 \/ Z9 \/ ZA by
XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8) \/ Z9 \/ ZA by
XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9) \/ ZA by
XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA) by
XBOOLE_1:4;
A14: now
assume
A15: Y in Z1;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9 such that
A16: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 &
Y7 in Y8 & Y8 in Y9 and
A17: Y9 in Y and
A18: Y1 meets X by A1;
Y in union X by A1,A15;
then Y9 in union union X by A17,TARSKI:def 4;
then Y9 in Z2 by A2,A16,A18;
then Y9 in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
then Y meets X \/ Z1 \/ Z2 by A17,XBOOLE_0:3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_1:70;
hence contradiction by A12,XBOOLE_1:70;
end;
assume
A19: not thesis;
now
assume
A20: Y in X;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA such that
A21: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 &
Y7 in Y8 & Y8 in Y9 & Y9 in YA and
A22: YA in Y and
A23: not Y1 misses X by A19;
YA in union X by A20,A22,TARSKI:def 4;
then YA in Z1 by A1,A21,A23;
then YA in X \/ Z1 by XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by A22,XBOOLE_0:3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_1:70;
hence contradiction by A12,XBOOLE_1:70;
end;
then Y in Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA by A13,A11
,XBOOLE_0:def 3;
then Y in Z1 \/ (Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA by
XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA by
XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5) \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA by
XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6) \/ Z7 \/ Z8 \/ Z9 \/ ZA by
XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7) \/ Z8 \/ Z9 \/ ZA by
XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8) \/ Z9 \/ ZA by
XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9) \/ ZA by
XBOOLE_1:4;
then Y in Z1 \/ (Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA) by
XBOOLE_1:4;
then Y in Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA by A14,
XBOOLE_0:def 3;
then Y in Z2 \/ (Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA by XBOOLE_1:4;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5) \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA by XBOOLE_1:4;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5 \/ Z6) \/ Z7 \/ Z8 \/ Z9 \/ ZA by XBOOLE_1:4;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7) \/ Z8 \/ Z9 \/ ZA by XBOOLE_1:4;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8) \/ Z9 \/ ZA by XBOOLE_1:4;
then Y in Z2 \/ (Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9) \/ ZA by XBOOLE_1:4;
then
A24: Y in Z2 \/ (Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA) by XBOOLE_1:4;
A25: now
assume
A26: Y in Z3;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7 such that
A27: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 and
A28: Y7 in Y and
A29: Y1 meets X by A6;
Y in union union union X by A6,A26;
then Y7 in union union union union X by A28,TARSKI:def 4;
then Y7 in Z4 by A8,A27,A29;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y7 in V by XBOOLE_0:def 3;
hence contradiction by A12,A28,XBOOLE_0:3;
end;
A30: now
assume
A31: Y in Z4;
then consider Y1,Y2,Y3,Y4,Y5,Y6 such that
A32: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 and
A33: Y6 in Y and
A34: Y1 meets X by A8;
Y in union union union union X by A8,A31;
then Y6 in union union union union union X by A33,TARSKI:def 4;
then Y6 in Z5 by A3,A32,A34;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y6 in V by XBOOLE_0:def 3;
hence contradiction by A12,A33,XBOOLE_0:3;
end;
A35: now
assume
A36: Y in Z5;
then consider Y1,Y2,Y3,Y4,Y5 such that
A37: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 and
A38: Y5 in Y and
A39: Y1 meets X by A3;
Y in union union union union union X by A3,A36;
then Y5 in union union union union union union X by A38,TARSKI:def 4;
then Y5 in Z6 by A4,A37,A39;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y5 in V by XBOOLE_0:def 3;
hence contradiction by A12,A38,XBOOLE_0:3;
end;
now
assume
A40: Y in Z2;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8 such that
A41: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 &
Y7 in Y8 and
A42: Y8 in Y and
A43: Y1 meets X by A2;
Y in union union X by A2,A40;
then Y8 in union union union X by A42,TARSKI:def 4;
then Y8 in Z3 by A6,A41,A43;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y8 in V by XBOOLE_0:def 3;
hence contradiction by A12,A42,XBOOLE_0:3;
end;
then Y in Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA by A24,XBOOLE_0:def 3;
then Y in Z3 \/ (Z4 \/ Z5) \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA by XBOOLE_1:4;
then Y in Z3 \/ (Z4 \/ Z5 \/ Z6) \/ Z7 \/ Z8 \/ Z9 \/ ZA by XBOOLE_1:4;
then Y in Z3 \/ (Z4 \/ Z5 \/ Z6 \/ Z7) \/ Z8 \/ Z9 \/ ZA by XBOOLE_1:4;
then Y in Z3 \/ (Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8) \/ Z9 \/ ZA by XBOOLE_1:4;
then Y in Z3 \/ (Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9) \/ ZA by XBOOLE_1:4;
then Y in Z3 \/ (Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA) by XBOOLE_1:4;
then Y in Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA by A25,XBOOLE_0:def 3;
then Y in Z4 \/ (Z5 \/ Z6) \/ Z7 \/ Z8 \/ Z9 \/ ZA by XBOOLE_1:4;
then Y in Z4 \/ (Z5 \/ Z6 \/ Z7) \/ Z8 \/ Z9 \/ ZA by XBOOLE_1:4;
then Y in Z4 \/ (Z5 \/ Z6 \/ Z7 \/ Z8) \/ Z9 \/ ZA by XBOOLE_1:4;
then Y in Z4 \/ (Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9) \/ ZA by XBOOLE_1:4;
then Y in Z4 \/ (Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA) by XBOOLE_1:4;
then Y in Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA by A30,XBOOLE_0:def 3;
then Y in Z5 \/ (Z6 \/ Z7) \/ Z8 \/ Z9 \/ ZA by XBOOLE_1:4;
then Y in Z5 \/ (Z6 \/ Z7 \/ Z8) \/ Z9 \/ ZA by XBOOLE_1:4;
then Y in Z5 \/ (Z6 \/ Z7 \/ Z8 \/ Z9) \/ ZA by XBOOLE_1:4;
then Y in Z5 \/ (Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA) by XBOOLE_1:4;
then Y in Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA by A35,XBOOLE_0:def 3;
then Y in Z6 \/ (Z7 \/ Z8) \/ Z9 \/ ZA by XBOOLE_1:4;
then Y in Z6 \/ (Z7 \/ Z8 \/ Z9) \/ ZA by XBOOLE_1:4;
then
A44: Y in Z6 \/ (Z7 \/ Z8 \/ Z9 \/ ZA) by XBOOLE_1:4;
A45: now
assume
A46: Y in Z7;
then consider Y1,Y2,Y3 such that
A47: Y1 in Y2 & Y2 in Y3 and
A48: Y3 in Y and
A49: Y1 meets X by A7;
Y in union union union union union union union X by A7,A46;
then
Y3 in union union union union union union union union X by A48,TARSKI:def 4;
then Y3 in Z8 by A10,A47,A49;
then Y3 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y3 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y3 in V by XBOOLE_0:def 3;
hence contradiction by A12,A48,XBOOLE_0:3;
end;
A50: now
assume
A51: Y in Z8;
then consider Y1,Y2 such that
A52: Y1 in Y2 and
A53: Y2 in Y and
A54: Y1 meets X by A10;
Y in union union union union union union union union X by A10,A51;
then Y2 in union union union union union union union union union X by A53,
TARSKI:def 4;
then Y2 in Z9 by A9,A52,A54;
then Y2 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y2 in V by XBOOLE_0:def 3;
hence contradiction by A12,A53,XBOOLE_0:3;
end;
A55: now
assume
A56: Y in Z9;
then consider Y1 such that
A57: Y1 in Y and
A58: Y1 meets X by A9;
Y in union union union union union union union union union X by A9,A56;
then Y1 in union union union union union union union union union union X
by A57,TARSKI:def 4;
then Y1 in ZA by A5,A58;
then Y1 in V by XBOOLE_0:def 3;
hence contradiction by A12,A57,XBOOLE_0:3;
end;
now
assume
A59: Y in Z6;
then consider Y1,Y2,Y3,Y4 such that
A60: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 and
A61: Y4 in Y and
A62: Y1 meets X by A4;
Y in union union union union union union X by A4,A59;
then Y4 in union union union union union union union X by A61,TARSKI:def 4;
then Y4 in Z7 by A7,A60,A62;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y4 in V by XBOOLE_0:def 3;
hence contradiction by A12,A61,XBOOLE_0:3;
end;
then Y in Z7 \/ Z8 \/ Z9 \/ ZA by A44,XBOOLE_0:def 3;
then Y in Z7 \/ (Z8 \/ Z9) \/ ZA by XBOOLE_1:4;
then Y in Z7 \/ (Z8 \/ Z9 \/ ZA) by XBOOLE_1:4;
then Y in Z8 \/ Z9 \/ ZA by A45,XBOOLE_0:def 3;
then Y in Z8 \/ (Z9 \/ ZA) by XBOOLE_1:4;
then Y in Z9 \/ ZA by A50,XBOOLE_0:def 3;
then Y in ZA by A55,XBOOLE_0:def 3;
then Y meets X by A5;
hence contradiction by A13,A12,XBOOLE_1:70;
end;
theorem Th77:
X <> {} implies ex Y st Y in X & for Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,
YA,YB st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7
in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in Y holds Y1 misses X
proof
defpred P1[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA st Y1 in Y2 & Y2 in
Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA
& YA in $1 & Y1 meets X;
consider Z1 such that
A1: for Y holds Y in Z1 iff Y in union X & P1[Y] from XBOOLE_0:sch 1;
defpred P11[set] means $1 meets X;
defpred P10[set] means ex Y1 st Y1 in $1 & Y1 meets X;
defpred P9[set] means ex Y1,Y2 st Y1 in Y2 & Y2 in $1 & Y1 meets X;
defpred P8[set] means ex Y1,Y2,Y3 st Y1 in Y2 & Y2 in Y3 & Y3 in $1 & Y1
meets X;
defpred P7[set] means ex Y1,Y2,Y3,Y4 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4
in $1 & Y1 meets X;
defpred P6[set] means ex Y1,Y2,Y3,Y4,Y5 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 &
Y4 in Y5 & Y5 in $1 & Y1 meets X;
defpred P5[set] means ex Y1,Y2,Y3,Y4,Y5,Y6 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4
& Y4 in Y5 & Y5 in Y6 & Y6 in $1 & Y1 meets X;
defpred P4[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7 st Y1 in Y2 & Y2 in Y3 & Y3 in
Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in $1 & Y1 meets X;
defpred P3[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8 st Y1 in Y2 & Y2 in Y3 & Y3
in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in $1 & Y1 meets X;
defpred P2[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9 st Y1 in Y2 & Y2 in Y3 &
Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in $1 & Y1
meets X;
consider Z2 such that
A2: for Y holds Y in Z2 iff Y in union union X & P2[Y] from XBOOLE_0:sch
1;
consider Z9 such that
A3: for Y holds Y in Z9 iff Y in union union union union union union
union union union X & P9[Y] from XBOOLE_0:sch 1;
consider Z8 such that
A4: for Y holds Y in Z8 iff Y in union union union union union union
union union X & P8[Y] from XBOOLE_0:sch 1;
consider ZB such that
A5: for Y holds Y in ZB iff Y in union union union union union union
union union union union (union X) & P11[Y] from XBOOLE_0:sch 1;
consider Z3 such that
A6: for Y holds Y in Z3 iff Y in union union union X & P3[Y] from
XBOOLE_0:sch 1;
consider Z7 such that
A7: for Y holds Y in Z7 iff Y in union union union union union union
union X & P7[Y] from XBOOLE_0:sch 1;
consider Z6 such that
A8: for Y holds Y in Z6 iff Y in union union union union union union X &
P6[Y] from XBOOLE_0:sch 1;
consider ZA such that
A9: for Y holds Y in ZA iff Y in union union union union union union
union union union union X & P10[Y] from XBOOLE_0:sch 1;
consider Z5 such that
A10: for Y holds Y in Z5 iff Y in union union union union union X & P5[Y]
from XBOOLE_0:sch 1;
consider Z4 such that
A11: for Y holds Y in Z4 iff Y in union union union union X & P4[Y] from
XBOOLE_0:sch 1;
set V = (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB;
A12: V = X \/ (Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB
by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB
by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5) \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB
by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6) \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB
by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7) \/ Z8 \/ Z9 \/ ZA \/ ZB
by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8) \/ Z9 \/ ZA \/ ZB
by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9) \/ ZA \/ ZB
by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA) \/ ZB
by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB)
by XBOOLE_1:4;
assume X <> {};
then consider Y such that
A13: Y in V and
A14: Y misses V by MCART_1:1;
Y in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA or
Y in ZB by A13,XBOOLE_0:def 3;
then Y in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 or Y in
ZA or Y in ZB by XBOOLE_0:def 3;
then Y in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 or Y in Z9 or
Y in ZA or Y in ZB by XBOOLE_0:def 3;
then Y in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 or Y in Z8 or Y in
Z9 or Y in ZA or Y in ZB by XBOOLE_0:def 3;
then
Y in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 or Y in Z7 or Y in Z8 or Y
in Z9 or Y in ZA or Y in ZB by XBOOLE_0:def 3;
then
Y in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 or Y in Z6 or Y in Z7 or Y in Z8
or Y in Z9 or Y in ZA or Y in ZB by XBOOLE_0:def 3;
then Y in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 or Y in Z5 or Y in Z6 or Y in Z7 or Y
in Z8 or Y in Z9 or Y in ZA or Y in ZB by XBOOLE_0:def 3;
then Y in (X \/ Z1) \/ Z2 \/ Z3 or Y in Z4 or Y in Z5 or Y in Z6 or Y in Z7
or Y in Z8 or Y in Z9 or Y in ZA or Y in ZB by XBOOLE_0:def 3;
then Y in (X \/ Z1) \/ Z2 or Y in Z3 or Y in Z4 or Y in Z5 or Y in Z6 or Y
in Z7 or Y in Z8 or Y in Z9 or Y in ZA or Y in ZB by XBOOLE_0:def 3;
then
A15: Y in X \/ Z1 or Y in Z2 or Y in Z3 or Y in Z4 or Y in Z5 or Y in Z6 or
Y in Z7 or Y in Z8 or Y in Z9 or Y in ZA or Y in ZB by XBOOLE_0:def 3;
assume
A16: not thesis;
per cases by A15,XBOOLE_0:def 3;
suppose
A17: Y in X;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB such that
A18: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 &
Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB and
A19: YB in Y and
A20: not Y1 misses X by A16;
YB in union X by A17,A19,TARSKI:def 4;
then YB in Z1 by A1,A18,A20;
then YB in X \/ Z1 by XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by A19,XBOOLE_0:3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_1:70;
then
Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_1:70;
hence contradiction by A14,XBOOLE_1:70;
end;
suppose
A21: Y in Z1;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA such that
A22: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 &
Y7 in Y8 & Y8 in Y9 & Y9 in YA and
A23: YA in Y and
A24: Y1 meets X by A1;
Y in union X by A1,A21;
then YA in union union X by A23,TARSKI:def 4;
then YA in Z2 by A2,A22,A24;
then YA in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
then Y meets X \/ Z1 \/ Z2 by A23,XBOOLE_0:3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_1:70;
then
Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_1:70;
hence contradiction by A14,XBOOLE_1:70;
end;
suppose
A25: Y in Z2;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9 such that
A26: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 &
Y7 in Y8 & Y8 in Y9 and
A27: Y9 in Y and
A28: Y1 meets X by A2;
Y in union union X by A2,A25;
then Y9 in union union union X by A27,TARSKI:def 4;
then Y9 in Z3 by A6,A26,A28;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y9 in V by XBOOLE_0:def 3;
hence contradiction by A14,A27,XBOOLE_0:3;
end;
suppose
A29: Y in Z3;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8 such that
A30: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7
& Y7 in Y8 and
A31: Y8 in Y and
A32: Y1 meets X by A6;
Y in union union union X by A6,A29;
then Y8 in union union union union X by A31,TARSKI:def 4;
then Y8 in Z4 by A11,A30,A32;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y8 in V by XBOOLE_0:def 3;
hence contradiction by A14,A31,XBOOLE_0:3;
end;
suppose
A33: Y in Z4;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7 such that
A34: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 and
A35: Y7 in Y and
A36: Y1 meets X by A11;
Y in union union union union X by A11,A33;
then Y7 in union union union union union X by A35,TARSKI:def 4;
then Y7 in Z5 by A10,A34,A36;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y7 in V by XBOOLE_0:def 3;
hence contradiction by A14,A35,XBOOLE_0:3;
end;
suppose
A37: Y in Z5;
then consider Y1,Y2,Y3,Y4,Y5,Y6 such that
A38: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 and
A39: Y6 in Y and
A40: Y1 meets X by A10;
Y in union union union union union X by A10,A37;
then Y6 in union union union union union union X by A39,TARSKI:def 4;
then Y6 in Z6 by A8,A38,A40;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y6 in V by XBOOLE_0:def 3;
hence contradiction by A14,A39,XBOOLE_0:3;
end;
suppose
A41: Y in Z6;
then consider Y1,Y2,Y3,Y4,Y5 such that
A42: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 and
A43: Y5 in Y and
A44: Y1 meets X by A8;
Y in union union union union union union X by A8,A41;
then Y5 in union union union union union union union X by A43,TARSKI:def 4;
then Y5 in Z7 by A7,A42,A44;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y5 in V by XBOOLE_0:def 3;
hence contradiction by A14,A43,XBOOLE_0:3;
end;
suppose
A45: Y in Z7;
then consider Y1,Y2,Y3,Y4 such that
A46: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 and
A47: Y4 in Y and
A48: Y1 meets X by A7;
Y in union union union union union union union X by A7,A45;
then
Y4 in union union union union union union union union X by A47,TARSKI:def 4;
then Y4 in Z8 by A4,A46,A48;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y4 in V by XBOOLE_0:def 3;
hence contradiction by A14,A47,XBOOLE_0:3;
end;
suppose
A49: Y in Z8;
then consider Y1,Y2,Y3 such that
A50: Y1 in Y2 & Y2 in Y3 and
A51: Y3 in Y and
A52: Y1 meets X by A4;
Y in union union union union union union union union X by A4,A49;
then Y3 in union union union union union union union union union X by A51,
TARSKI:def 4;
then Y3 in Z9 by A3,A50,A52;
then Y3 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y3 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y3 in V by XBOOLE_0:def 3;
hence contradiction by A14,A51,XBOOLE_0:3;
end;
suppose
A53: Y in Z9;
then consider Y1,Y2 such that
A54: Y1 in Y2 and
A55: Y2 in Y and
A56: Y1 meets X by A3;
Y in union union union union union union union union union X by A3,A53;
then Y2 in union union union union union union union union union union X
by A55,TARSKI:def 4;
then Y2 in ZA by A9,A54,A56;
then Y2 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y2 in V by XBOOLE_0:def 3;
hence contradiction by A14,A55,XBOOLE_0:3;
end;
suppose
A57: Y in ZA;
then consider Y1 such that
A58: Y1 in Y and
A59: Y1 meets X by A9;
Y in union union union union union union union union union union X
by A9,A57;
then Y1 in union union union union union union union union union union (
union X) by A58,TARSKI:def 4;
then Y1 in ZB by A5,A59;
then Y1 in V by XBOOLE_0:def 3;
hence contradiction by A14,A58,XBOOLE_0:3;
end;
suppose
Y in ZB;
then Y meets X by A5;
hence contradiction by A12,A14,XBOOLE_1:70;
end;
end;
::
:: Tuples for n=7
::
definition
let x1,x2,x3,x4,x5,x6,x7;
func [x1,x2,x3,x4,x5,x6,x7] equals
[[x1,x2,x3,x4,x5,x6],x7];
correctness;
end;
theorem Th78:
[x1,x2,x3,x4,x5,x6,x7] = [[[[[[x1,x2],x3],x4],x5],x6],x7]
proof
thus [x1,x2,x3,x4,x5,x6,x7] = [[[[[x1,x2,x3],x4],x5],x6],x7] by MCART_1:def 4
.= [[[[[[x1,x2],x3],x4],x5],x6],x7] by MCART_1:def 3;
end;
theorem
[x1,x2,x3,x4,x5,x6,x7] = [[x1,x2,x3,x4,x5],x6,x7] by MCART_1:def 3;
theorem
[x1,x2,x3,x4,x5,x6,x7] = [[x1,x2,x3,x4],x5,x6,x7] by MCART_1:27;
theorem
[x1,x2,x3,x4,x5,x6,x7] = [[x1,x2,x3],x4,x5,x6,x7]
proof
thus [x1,x2,x3,x4,x5,x6,x7] = [[[[[[x1,x2],x3],x4],x5],x6],x7] by Th78
.= [[[x1,x2],x3],x4,x5,x6,x7] by Th3
.= [[x1,x2,x3],x4,x5,x6,x7] by MCART_1:def 3;
end;
theorem
[x1,x2,x3,x4,x5,x6,x7] = [[x1,x2],x3,x4,x5,x6,x7] by Th6;
theorem Th83:
[x1,x2,x3,x4,x5,x6,x7] = [y1,y2,y3,y4,y5,y6,y7] implies x1 = y1
& x2 = y2 & x3 = y3 & x4 = y4 & x5 = y5 & x6 = y6 & x7 = y7
proof
assume
A1: [x1,x2,x3,x4,x5,x6,x7] = [y1,y2,y3,y4,y5,y6,y7];
then [x1,x2,x3,x4,x5,x6] = [y1,y2,y3,y4,y5,y6] by ZFMISC_1:27;
hence thesis by A1,Th45,ZFMISC_1:27;
end;
theorem Th84:
X <> {} implies ex x being set st x in X & not ex x1,x2,x3,x4,x5
,x6,x7 st (x1 in X or x2 in X) & x = [x1,x2,x3,x4,x5,x6,x7]
proof
assume X <> {};
then consider Y such that
A1: Y in X and
A2: for Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB st Y1 in Y2 & Y2 in Y3 & Y3 in
Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB
& YB in Y holds Y1 misses X by Th77;
take x = Y;
thus x in X by A1;
given x1,x2,x3,x4,x5,x6,x7 such that
A3: x1 in X or x2 in X and
A4: x = [x1,x2,x3,x4,x5,x6,x7];
set Y1 = { x1, x2 }, Y2 = { Y1, {x1} }, Y3 = { Y2, x3 }, Y4 = { Y3, {Y2} },
Y5 = { Y4, x4 }, Y6 = { Y5, {Y4} }, Y7 = { Y6, x5 }, Y8 = { Y7, {Y6} }, Y9 = {
Y8, x6 }, YA = { Y9, {Y8} }, YB = { YA, x7 };
A5: Y3 in Y4 & Y4 in Y5 by TARSKI:def 2;
x1 in Y1 & x2 in Y1 by TARSKI:def 2;
then
A6: not Y1 misses X by A3,XBOOLE_0:3;
A7: Y7 in Y8 & Y8 in Y9 by TARSKI:def 2;
Y = [[[[[[x1,x2],x3],x4],x5],x6],x7] by A4,Th78
.= [[[[[ Y2,x3],x4],x5],x6],x7 ] by TARSKI:def 5
.= [[[[ Y4,x4],x5],x6],x7 ] by TARSKI:def 5
.= [[[ Y6,x5 ],x6],x7 ] by TARSKI:def 5
.= [[ Y8,x6],x7 ] by TARSKI:def 5
.= [ YA,x7 ] by TARSKI:def 5
.= { YB, { YA } } by TARSKI:def 5;
then
A8: YB in Y by TARSKI:def 2;
A9: Y5 in Y6 & Y6 in Y7 by TARSKI:def 2;
A10: Y9 in YA & YA in YB by TARSKI:def 2;
Y1 in Y2 & Y2 in Y3 by TARSKI:def 2;
hence contradiction by A2,A6,A5,A9,A7,A10,A8;
end;
::
:: Cartesian products of seven sets
::
definition
let X1,X2,X3,X4,X5,X6,X7;
func [:X1,X2,X3,X4,X5,X6,X7:] -> set equals
[:[: X1,X2,X3,X4,X5,X6 :],X7 :];
correctness;
end;
theorem Th85:
[:X1,X2,X3,X4,X5,X6,X7:] = [:[:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6 :],X7:]
proof
thus [:X1,X2,X3,X4,X5,X6,X7:] = [:[:[:[:[:X1,X2,X3:],X4:],X5:],X6:],X7:] by
ZFMISC_1:def 4
.= [:[:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:],X7:] by ZFMISC_1:def 3;
end;
theorem
[:X1,X2,X3,X4,X5,X6,X7:] = [:[:X1,X2,X3,X4,X5:],X6,X7:] by ZFMISC_1:def 3;
theorem
[:X1,X2,X3,X4,X5,X6,X7:] = [:[:X1,X2,X3,X4:],X5,X6,X7:] by MCART_1:49;
theorem
[:X1,X2,X3,X4,X5,X6,X7:] = [:[:X1,X2,X3:],X4,X5,X6,X7:]
proof
thus [:X1,X2,X3,X4,X5,X6,X7:] = [:[:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:],X7:]
by Th85
.= [:[:[:X1,X2:],X3:],X4,X5,X6,X7:] by Th9
.= [:[:X1,X2,X3:],X4,X5,X6,X7:] by ZFMISC_1:def 3;
end;
theorem
[:X1,X2,X3,X4,X5,X6,X7:] = [:[:X1,X2:],X3,X4,X5,X6,X7:] by Th12;
theorem Th90:
X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {}
& X7 <> {} iff [:X1,X2,X3,X4,X5,X6,X7:] <> {}
proof
X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} iff [:X1
,X2,X3,X4,X5,X6:] <> {} by Th51;
hence thesis by ZFMISC_1:90;
end;
theorem Th91:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<> {}
implies ( [:X1,X2,X3,X4,X5,X6,X7:] = [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:] implies X1=Y1 &
X2=Y2 & X3=Y3 & X4=Y4 & X5=Y5 & X6=Y6 & X7=Y7 )
proof
assume
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{};
then
A2: [:X1,X2,X3,X4,X5,X6:] <> {} by Th51;
assume
A3: X7<>{};
assume
A4: [:X1,X2,X3,X4,X5,X6,X7:] = [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:];
then [:X1,X2,X3,X4,X5,X6:] = [:Y1,Y2,Y3,Y4,Y5,Y6:] by A2,A3,ZFMISC_1:110;
hence thesis by A1,A2,A3,A4,Th52,ZFMISC_1:110;
end;
theorem
[:X1,X2,X3,X4,X5,X6,X7:]<>{} & [:X1,X2,X3,X4,X5,X6,X7:] = [:Y1,Y2,Y3,
Y4,Y5,Y6,Y7:] implies X1=Y1 & X2=Y2 & X3=Y3 & X4=Y4 & X5=Y5 & X6=Y6 & X7=Y7
proof
assume
A1: [:X1,X2,X3,X4,X5,X6,X7:]<>{};
then
A2: X3<>{} & X4<>{} by Th90;
A3: X7<>{} by A1,Th90;
A4: X5<>{} & X6<>{} by A1,Th90;
X1<>{} & X2<>{} by A1,Th90;
hence thesis by A2,A4,A3,Th91;
end;
theorem
[:X,X,X,X,X,X,X:] = [:Y,Y,Y,Y,Y,Y,Y:] implies X = Y
proof
assume [:X,X,X,X,X,X,X:] = [:Y,Y,Y,Y,Y,Y,Y:];
then X<>{} or Y<>{} implies thesis by Th91;
hence thesis;
end;
theorem Th94:
X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {}
& X7 <> {} implies for x being Element of [:X1,X2,X3,X4,X5,X6,X7:] ex xx1,xx2,
xx3,xx4,xx5,xx6,xx7 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7]
proof
assume that
A1: X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} and
A2: X7 <> {};
let x be Element of [:X1,X2,X3,X4,X5,X6,X7:];
reconsider x9=x as Element of [:[:X1,X2,X3,X4,X5,X6:],X7:];
[:X1,X2,X3,X4,X5,X6:] <> {} by A1,Th51;
then consider
x123456 being (Element of [:X1,X2,X3,X4,X5,X6:]), xx7 such that
A3: x9 = [x123456,xx7] by A2,Lm1;
consider xx1,xx2,xx3,xx4,xx5,xx6 such that
A4: x123456 = [xx1,xx2,xx3,xx4,xx5,xx6] by A1,Th55;
take xx1,xx2,xx3,xx4,xx5,xx6,xx7;
thus thesis by A3,A4;
end;
definition
let X1,X2,X3,X4,X5,X6,X7;
assume
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{};
let x be Element of [:X1,X2,X3,X4,X5,X6,X7:];
func x`1 -> Element of X1 means
:Def18:
x = [x1,x2,x3,x4,x5,x6,x7] implies it = x1;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A2: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th94;
take xx1;
thus thesis by A2,Th83;
end;
uniqueness
proof
let y,z be Element of X1;
assume
A3: x = [x1,x2,x3,x4,x5,x6,x7] implies y = x1;
assume
A4: x = [x1,x2,x3,x4,x5,x6,x7] implies z = x1;
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A5: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th94;
y = xx1 by A5,A3;
hence thesis by A5,A4;
end;
func x`2 -> Element of X2 means
:Def19:
x = [x1,x2,x3,x4,x5,x6,x7] implies it = x2;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A6: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th94;
take xx2;
thus thesis by A6,Th83;
end;
uniqueness
proof
let y,z be Element of X2;
assume
A7: x = [x1,x2,x3,x4,x5,x6,x7] implies y = x2;
assume
A8: x = [x1,x2,x3,x4,x5,x6,x7] implies z = x2;
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A9: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th94;
y = xx2 by A9,A7;
hence thesis by A9,A8;
end;
func x`3 -> Element of X3 means
:Def20:
x = [x1,x2,x3,x4,x5,x6,x7] implies it = x3;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A10: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th94;
take xx3;
thus thesis by A10,Th83;
end;
uniqueness
proof
let y,z be Element of X3;
assume
A11: x = [x1,x2,x3,x4,x5,x6,x7] implies y = x3;
assume
A12: x = [x1,x2,x3,x4,x5,x6,x7] implies z = x3;
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A13: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th94;
y = xx3 by A13,A11;
hence thesis by A13,A12;
end;
func x`4 -> Element of X4 means
:Def21:
x = [x1,x2,x3,x4,x5,x6,x7] implies it = x4;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A14: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th94;
take xx4;
thus thesis by A14,Th83;
end;
uniqueness
proof
let y,z be Element of X4;
assume
A15: x = [x1,x2,x3,x4,x5,x6,x7] implies y = x4;
assume
A16: x = [x1,x2,x3,x4,x5,x6,x7] implies z = x4;
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A17: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th94;
y = xx4 by A17,A15;
hence thesis by A17,A16;
end;
func x`5 -> Element of X5 means
:Def22:
x = [x1,x2,x3,x4,x5,x6,x7] implies it = x5;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A18: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th94;
take xx5;
thus thesis by A18,Th83;
end;
uniqueness
proof
let y,z be Element of X5;
assume
A19: x = [x1,x2,x3,x4,x5,x6,x7] implies y = x5;
assume
A20: x = [x1,x2,x3,x4,x5,x6,x7] implies z = x5;
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A21: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th94;
y = xx5 by A21,A19;
hence thesis by A21,A20;
end;
func x`6 -> Element of X6 means
:Def23:
x = [x1,x2,x3,x4,x5,x6,x7] implies it = x6;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A22: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th94;
take xx6;
thus thesis by A22,Th83;
end;
uniqueness
proof
let y,z be Element of X6;
assume
A23: x = [x1,x2,x3,x4,x5,x6,x7] implies y = x6;
assume
A24: x = [x1,x2,x3,x4,x5,x6,x7] implies z = x6;
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A25: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th94;
y = xx6 by A25,A23;
hence thesis by A25,A24;
end;
func x`7 -> Element of X7 means
:Def24:
x = [x1,x2,x3,x4,x5,x6,x7] implies it = x7;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A26: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th94;
take xx7;
thus thesis by A26,Th83;
end;
uniqueness
proof
let y,z be Element of X7;
assume
A27: x = [x1,x2,x3,x4,x5,x6,x7] implies y = x7;
assume
A28: x = [x1,x2,x3,x4,x5,x6,x7] implies z = x7;
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A29: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th94;
y = xx7 by A29,A27;
hence thesis by A29,A28;
end;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} implies
for x being Element of [:X1,X2,X3,X4,X5,X6,X7:] for x1,x2,x3,x4,x5,x6,x7 st x =
[x1,x2,x3,x4,x5,x6,x7] holds x`1 = x1 & x`2 = x2 & x`3 = x3 & x`4 = x4 & x`5 =
x5 & x`6 = x6 & x`7 = x7 by Def18,Def19,Def20,Def21,Def22,Def23,Def24;
theorem Th96:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{}
implies for x being Element of [:X1,X2,X3,X4,X5,X6,X7:] holds x = [x`1,x`2,x`3,
x`4,x`5,x`6,x`7]
proof
assume
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{};
let x be Element of [:X1,X2,X3,X4,X5,X6,X7:];
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A2: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th94;
thus x = [x`1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,A2,Def18
.= [x`1,x`2,xx3,xx4,xx5,xx6,xx7] by A1,A2,Def19
.= [x`1,x`2,x`3,xx4,xx5,xx6,xx7] by A1,A2,Def20
.= [x`1,x`2,x`3,x`4,xx5,xx6,xx7] by A1,A2,Def21
.= [x`1,x`2,x`3,x`4,x`5,xx6,xx7] by A1,A2,Def22
.= [x`1,x`2,x`3,x`4,x`5,x`6,xx7] by A1,A2,Def23
.= [x`1,x`2,x`3,x`4,x`5,x`6,x`7] by A1,A2,Def24;
end;
theorem Th97:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{}
implies for x being Element of [:X1,X2,X3,X4,X5,X6,X7:] holds x`1 = (x qua set)
`1`1`1`1`1`1 & x`2 = (x qua set)`1`1`1`1`1`2 & x`3 = (x qua set)`1`1`1`1`2 & x
`4 = (x qua set)`1`1`1`2 & x`5 = (x qua set)`1`1`2 & x`6 = (x qua set)`1`2 & x
`7 = (x qua set)`2
proof
assume
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{};
let x be Element of [:X1,X2,X3,X4,X5,X6,X7:];
thus x`1 = [ x`1, x`2]`1 by MCART_1:7
.= [[x`1, x`2],x`3]`1`1 by MCART_1:7
.= [ x`1, x`2 ,x`3]`1`1 by MCART_1:def 3
.= [[x`1, x`2 ,x`3],x`4]`1`1`1 by MCART_1:7
.= [ x`1, x`2 ,x`3 ,x`4]`1`1`1 by MCART_1:def 4
.= [[x`1, x`2 ,x`3 ,x`4], x`5]`1`1`1`1 by MCART_1:7
.= [[x`1, x`2 ,x`3 ,x`4, x`5],x`6]`1`1`1`1`1 by MCART_1:7
.= [ x`1, x`2 ,x`3 ,x`4 ,x`5, x`6, x`7]`1`1`1`1`1`1 by MCART_1:7
.= (x qua set)`1`1`1`1`1`1 by A1,Th96;
thus x`2 = [ x`1, x`2]`2 by MCART_1:7
.= [[x`1, x`2],x`3]`1`2 by MCART_1:7
.= [ x`1, x`2, x`3]`1`2 by MCART_1:def 3
.= [[x`1, x`2, x`3],x`4]`1`1`2 by MCART_1:7
.= [ x`1, x`2, x`3, x`4]`1`1`2 by MCART_1:def 4
.= [[x`1, x`2, x`3, x`4 ], x`5]`1`1`1`2 by MCART_1:7
.= [[x`1, x`2, x`3, x`4, x`5],x`6]`1`1`1`1`2 by MCART_1:7
.= [ x`1, x`2, x`3, x`4, x`5, x`6, x`7]`1`1`1`1`1`2 by MCART_1:7
.= (x qua set)`1`1`1`1`1`2 by A1,Th96;
thus x`3 = [[x`1, x`2],x`3]`2 by MCART_1:7
.= [ x`1, x`2, x`3]`2 by MCART_1:def 3
.= [[x`1, x`2, x`3],x`4]`1`2 by MCART_1:7
.= [ x`1, x`2, x`3, x`4]`1`2 by MCART_1:def 4
.= [[x`1, x`2, x`3, x`4],x`5]`1`1`2 by MCART_1:7
.= [[x`1, x`2, x`3, x`4, x`5],x`6]`1`1`1`2 by MCART_1:7
.= [ x`1, x`2, x`3, x`4, x`5, x`6, x`7]`1`1`1`1`2 by MCART_1:7
.= (x qua set)`1`1`1`1`2 by A1,Th96;
thus x`4 = [[x`1,x`2,x`3],x`4]`2 by MCART_1:7
.= [ x`1,x`2,x`3, x`4]`2 by MCART_1:def 4
.= [[x`1,x`2,x`3, x`4],x`5]`1`2 by MCART_1:7
.= [[x`1,x`2,x`3, x`4, x`5],x`6]`1`1`2 by MCART_1:7
.= [ x`1,x`2,x`3, x`4, x`5, x`6, x`7]`1`1`1`2 by MCART_1:7
.= (x qua set)`1`1`1`2 by A1,Th96;
thus x`5 = [[x`1,x`2,x`3,x`4],x`5]`2 by MCART_1:7
.= [[x`1,x`2,x`3,x`4,x`5],x`6]`1`2 by MCART_1:7
.= [ x`1,x`2,x`3,x`4,x`5, x`6,x`7]`1`1`2 by MCART_1:7
.= (x qua set)`1`1`2 by A1,Th96;
thus x`6 = [[x`1,x`2,x`3,x`4,x`5],x`6]`2 by MCART_1:7
.= [ x`1,x`2,x`3,x`4,x`5,x`6, x`7]`1`2 by MCART_1:7
.= (x qua set)`1`2 by A1,Th96;
thus x`7 = [ x`1,x`2,x`3,x`4,x`5,x`6, x`7]`2 by MCART_1:7
.= (x qua set)`2 by A1,Th96;
end;
theorem Th98:
X1 c= [:X1,X2,X3,X4,X5,X6,X7:] or X1 c= [:X2,X3,X4,X5,X6,X7,X1:]
or X1 c= [:X3,X4,X5,X6,X7,X1,X2:] or X1 c= [:X4,X5,X6,X7,X1,X2,X3:] or X1 c= [:
X5,X6,X7,X1,X2,X3,X4:] or X1 c= [:X6,X7,X1,X2,X3,X4,X5:] or X1 c= [:X7,X1,X2,X3
,X4,X5,X6:] implies X1 = {}
proof
assume that
A1: X1 c= [:X1,X2,X3,X4,X5,X6,X7:] or X1 c= [:X2,X3,X4,X5,X6,X7,X1:] or
X1 c= [:X3,X4,X5,X6,X7,X1,X2:] or X1 c= [:X4,X5,X6,X7,X1,X2,X3:] or X1 c= [:X5,
X6,X7,X1,X2,X3,X4:] or X1 c= [:X6,X7,X1,X2,X3,X4,X5:] or X1 c= [:X7,X1,X2,X3,X4
,X5,X6:] and
A2: X1 <> {};
A3: [:X1,X2,X3,X4,X5,X6,X7:]<>{} or [:X2,X3,X4,X5,X6,X7,X1:]<>{} or [:X3,X4,
X5,X6,X7,X1,X2:]<>{} or [:X4,X5,X6,X7,X1,X2,X3:]<>{} or [:X5,X6,X7,X1,X2,X3,X4
:]<>{} or [:X6,X7,X1,X2,X3,X4,X5:]<>{} or [:X7,X1,X2,X3,X4,X5,X6:]<>{} by A1,A2
,XBOOLE_1:3;
then
A4: X4<>{} & X5<>{} by Th90;
A5: X6<>{} & X7<>{} by A3,Th90;
A6: X2<>{} & X3<>{} by A3,Th90;
now
per cases by A1;
suppose
A7: X1 c= [:X1,X2,X3,X4,X5,X6,X7:];
consider y such that
A8: y in X1 and
A9: for x1,x2,x3,x4,x5,x6,x7 st x1 in X1 or x2 in X1 holds y <> [x1
,x2,x3,x4,x5,x6,x7] by A2,Th84;
reconsider y as Element of [:X1,X2,X3,X4,X5,X6,X7:] by A7,A8;
y = [y`1,y`2,y`3,y`4,y`5,y`6,y`7] by A2,A6,A4,A5,Th96;
hence contradiction by A2,A9;
end;
suppose
X1 c= [:X2,X3,X4,X5,X6,X7,X1:];
then X1 c= [:[:X2,X3:],X4,X5,X6,X7,X1:] by Th12;
hence thesis by A2,Th59;
end;
suppose
X1 c= [:X3,X4,X5,X6,X7,X1,X2:];
then X1 c= [:[:X3,X4:],X5,X6,X7,X1,X2:] by Th12;
hence thesis by A2,Th59;
end;
suppose
X1 c= [:X4,X5,X6,X7,X1,X2,X3:];
then X1 c= [:[:X4,X5:],X6,X7,X1,X2,X3:] by Th12;
hence thesis by A2,Th59;
end;
suppose
X1 c= [:X5,X6,X7,X1,X2,X3,X4:];
then X1 c= [:[:X5,X6:],X7,X1,X2,X3,X4:] by Th12;
hence thesis by A2,Th59;
end;
suppose
X1 c= [:X6,X7,X1,X2,X3,X4,X5:];
then X1 c= [:[:X6,X7:],X1,X2,X3,X4,X5:] by Th12;
hence thesis by A2,Th59;
end;
suppose
A10: X1 c= [:X7,X1,X2,X3,X4,X5,X6:];
consider y such that
A11: y in X1 and
A12: for x1,x2,x3,x4,x5,x6,x7 st x1 in X1 or x2 in X1 holds y <> [x1
,x2,x3,x4,x5,x6,x7] by A2,Th84;
reconsider y as Element of [:X7,X1,X2,X3,X4,X5,X6:] by A10,A11;
y = [y`1,y`2,y`3,y`4,y`5,y`6,y`7] by A2,A6,A4,A5,Th96;
hence thesis by A2,A12;
end;
end;
hence contradiction;
end;
theorem Th99:
[:X1,X2,X3,X4,X5,X6,X7:] meets [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:] implies
X1 meets Y1 & X2 meets Y2 & X3 meets Y3 & X4 meets Y4 & X5 meets Y5 & X6 meets
Y6 & X7 meets Y7
proof
assume
A1: [:X1,X2,X3,X4,X5,X6,X7:] meets [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:];
[:[:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:],X7:] = [:X1,X2,X3,X4,X5,X6,X7:]
& [: [:[:[:[:[:Y1,Y2:],Y3:],Y4:],Y5:],Y6:],Y7:] = [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:]
by Th85;
then
A2: [:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:] meets [:[:[:[:[:Y1,Y2:],Y3:],Y4:],
Y5:],Y6:] by A1,ZFMISC_1:104;
then
A3: [:[:[:[:X1,X2:],X3:],X4:],X5:] meets [:[:[:[:Y1,Y2:],Y3:],Y4:],Y5:] by
ZFMISC_1:104;
then
A4: [:[:[:X1,X2:],X3:],X4:] meets [:[:[:Y1,Y2:],Y3:],Y4:] by ZFMISC_1:104;
then
A5: [:[:X1,X2:],X3:] meets [:[:Y1,Y2:],Y3:] by ZFMISC_1:104;
then [:X1,X2:] meets [:Y1,Y2:] by ZFMISC_1:104;
hence thesis by A1,A2,A3,A4,A5,ZFMISC_1:104;
end;
theorem Th100:
[:{x1},{x2},{x3},{x4},{x5},{x6},{x7}:] = { [x1,x2,x3,x4,x5,x6, x7] }
proof
thus [:{x1},{x2},{x3},{x4},{x5},{x6},{x7}:] = [:[:{x1},{x2}:],{x3},{x4},{x5}
,{x6},{x7}:] by Th12
.= [:{[x1,x2]}, {x3},{x4},{x5},{x6},{x7}:] by ZFMISC_1:29
.= { [[x1,x2], x3, x4, x5, x6, x7]} by Th61
.= { [x1,x2,x3,x4,x5,x6,x7] } by Th6;
end;
:: 7 - Tuples
reserve x for Element of [:X1,X2,X3,X4,X5,X6,X7:];
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} implies
for x1,x2,x3,x4,x5,x6,x7 st x = [x1,x2,x3,x4,x5,x6,x7] holds x`1 = x1 & x`2 =
x2 & x`3 = x3 & x`4 = x4 & x`5 = x5 & x`6 = x6 & x`7 = x7 by Def18,Def19,Def20
,Def21,Def22,Def23,Def24;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & (for
xx1,xx2,xx3,xx4,xx5,xx6,xx7 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds y1 = xx1
) implies y1 =x`1
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6,xx7 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7]
holds y1 = xx1;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7] by A1,Th96;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & (for
xx1,xx2,xx3,xx4,xx5,xx6,xx7 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds y2 = xx2
) implies y2 =x`2
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6,xx7 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7]
holds y2 = xx2;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7] by A1,Th96;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & (for
xx1,xx2,xx3,xx4,xx5,xx6,xx7 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds y3 = xx3
) implies y3 =x`3
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6,xx7 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7]
holds y3 = xx3;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7] by A1,Th96;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & (for
xx1,xx2,xx3,xx4,xx5,xx6,xx7 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds y4 = xx4
) implies y4 =x`4
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6,xx7 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7]
holds y4 = xx4;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7] by A1,Th96;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & (for
xx1,xx2,xx3,xx4,xx5,xx6,xx7 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds y5 = xx5
) implies y5 =x`5
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6,xx7 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7]
holds y5 = xx5;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7] by A1,Th96;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & (for
xx1,xx2,xx3,xx4,xx5,xx6,xx7 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds y6 = xx6
) implies y6 =x`6
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6,xx7 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7]
holds y6 = xx6;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7] by A1,Th96;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & (for
xx1,xx2,xx3,xx4,xx5,xx6,xx7 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] holds y7 = xx7
) implies y7 =x`7
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6,xx7 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7]
holds y7 = xx7;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7] by A1,Th96;
hence thesis by A2;
end;
theorem Th109:
y in [: X1,X2,X3,X4,X5,X6,X7 :] implies ex x1,x2,x3,x4,x5,x6,x7
st x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7 in X7 &
y = [x1,x2,x3,x4,x5,x6,x7]
proof
assume y in [: X1,X2,X3,X4,X5,X6,X7 :];
then consider x123456, x7 being set such that
A1: x123456 in [:X1,X2,X3,X4,X5,X6:] and
A2: x7 in X7 and
A3: y = [x123456,x7] by ZFMISC_1:def 2;
consider x1, x2, x3, x4, x5, x6 such that
A4: x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 and
A5: x123456 = [x1,x2,x3,x4,x5,x6] by A1,Th69;
y = [x1,x2,x3,x4,x5,x6,x7] by A3,A5;
hence thesis by A2,A4;
end;
theorem Th110:
[x1,x2,x3,x4,x5,x6,x7] in [: X1,X2,X3,X4,X5,X6,X7 :] iff x1 in
X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7 in X7
proof
A1: [x1,x2] in [:X1,X2:] iff x1 in X1 & x2 in X2 by ZFMISC_1:87;
[:X1,X2,X3,X4,X5,X6,X7:] = [:[:X1,X2:],X3,X4,X5,X6,X7:] & [x1,x2,x3,x4,
x5,x6,x7] = [[x1,x2],x3,x4,x5,x6,x7] by Th6,Th12;
hence thesis by A1,Th70;
end;
theorem
(for y holds y in Z iff ex x1,x2,x3,x4,x5,x6,x7 st x1 in X1 & x2 in X2
& x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7 in X7 & y = [x1,x2,x3,x4,x5,x6
,x7]) implies Z = [: X1,X2,X3,X4,X5,X6,X7 :]
proof
assume
A1: for y holds y in Z iff ex x1,x2,x3,x4,x5,x6,x7 st x1 in X1 & x2 in
X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7 in X7 & y = [x1,x2,x3,x4,x5
,x6,x7];
now
let y;
thus y in Z implies y in [:[:X1,X2,X3,X4,X5,X6:],X7:]
proof
assume y in Z;
then consider x1,x2,x3,x4,x5,x6,x7 such that
A2: x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 and
A3: x7 in X7 & y = [x1,x2,x3,x4,x5,x6,x7] by A1;
[x1,x2,x3,x4,x5,x6] in [:X1,X2,X3,X4,X5,X6:] by A2,Th70;
hence thesis by A3,ZFMISC_1:def 2;
end;
assume y in [:[:X1,X2,X3,X4,X5,X6:],X7:];
then consider x123456,x7 being set such that
A4: x123456 in [:X1,X2,X3,X4,X5,X6:] and
A5: x7 in X7 and
A6: y = [x123456,x7] by ZFMISC_1:def 2;
consider x1,x2,x3,x4,x5,x6 such that
A7: x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 and
A8: x123456 = [x1,x2,x3,x4,x5,x6] by A4,Th69;
y = [x1,x2,x3,x4,x5,x6,x7] by A6,A8;
hence y in Z by A1,A5,A7;
end;
hence thesis by TARSKI:1;
end;
theorem Th112:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} &
Y1<>{} & Y2<>{} & Y3<>{} & Y4<>{} & Y5<>{} & Y6<>{} & Y7<>{} implies for x
being (Element of [:X1,X2,X3,X4,X5,X6,X7:]), y being Element of [:Y1,Y2,Y3,Y4,
Y5,Y6,Y7:] holds x = y implies x`1 = y`1 & x`2 = y`2 & x`3 = y`3 & x`4 = y`4 &
x`5 = y`5 & x`6 = y`6 & x`7 = y`7
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} and
A2: Y1<>{} & Y2<>{} & Y3<>{} & Y4<>{} & Y5<>{} & Y6<>{} & Y7<>{};
let x be Element of [:X1,X2,X3,X4,X5,X6,X7:];
let y be Element of [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:];
assume
A3: x = y;
thus x`1 = (x qua set)`1`1`1`1`1`1 by A1,Th97
.= y`1 by A2,A3,Th97;
thus x`2 = (x qua set)`1`1`1`1`1`2 by A1,Th97
.= y`2 by A2,A3,Th97;
thus x`3 = (x qua set)`1`1`1`1`2 by A1,Th97
.= y`3 by A2,A3,Th97;
thus x`4 = (x qua set)`1`1`1`2 by A1,Th97
.= y`4 by A2,A3,Th97;
thus x`5 = (x qua set)`1`1`2 by A1,Th97
.= y`5 by A2,A3,Th97;
thus x`6 = (x qua set)`1`2 by A1,Th97
.= y`6 by A2,A3,Th97;
thus x`7 = (x qua set)`2 by A1,Th97
.= y`7 by A2,A3,Th97;
end;
theorem
for x being Element of [:X1,X2,X3,X4,X5,X6,X7:] st x in [:A1,A2,A3,A4,
A5,A6,A7:] holds x`1 in A1 & x`2 in A2 & x`3 in A3 & x`4 in A4 & x`5 in A5 & x
`6 in A6 & x`7 in A7
proof
let x be Element of [:X1,X2,X3,X4,X5,X6,X7:];
assume
A1: x in [:A1,A2,A3,A4,A5,A6,A7:];
then reconsider y = x as Element of [:A1,A2,A3,A4,A5,A6,A7:];
A2<>{} by A1,Th90;
then
A2: y`2 in A2;
A7<>{} by A1,Th90;
then
A3: y`7 in A7;
A4<>{} by A1,Th90;
then
A4: y`4 in A4;
A3<>{} by A1,Th90;
then
A5: y`3 in A3;
A6<>{} by A1,Th90;
then
A6: y`6 in A6;
A5<>{} by A1,Th90;
then
A7: y`5 in A5;
A1<>{} by A1,Th90;
then y`1 in A1;
hence thesis by A2,A5,A4,A7,A6,A3,Th112;
end;
theorem Th114:
X1 c= Y1 & X2 c= Y2 & X3 c= Y3 & X4 c= Y4 & X5 c= Y5 & X6 c= Y6
& X7 c= Y7 implies [:X1,X2,X3,X4,X5,X6,X7:] c= [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:]
proof
assume X1 c= Y1 & X2 c= Y2 & X3 c= Y3 & X4 c= Y4 & X5 c= Y5 & X6 c= Y6;
then
A1: [:X1,X2,X3,X4,X5,X6:] c= [:Y1,Y2,Y3,Y4,Y5,Y6:] by Th74;
assume X7 c= Y7;
hence thesis by A1,ZFMISC_1:96;
end;
theorem
[:A1,A2,A3,A4,A5,A6,A7:] is Subset of [:X1,X2,X3,X4,X5,X6,X7:] by Th114;
begin
theorem
X <> {} implies ex Y st Y in X & for Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB,
YC st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in
Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC & YC in Y holds Y1 misses X
proof
defpred P1[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB st Y1 in Y2 & Y2
in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in
YA & YA in YB & YB in $1 & Y1 meets X;
consider Z1 such that
A1: for Y holds Y in Z1 iff Y in union X & P1[Y] from XBOOLE_0:sch 1;
defpred P12[set] means $1 meets X;
defpred P11[set] means ex Y1 st Y1 in $1 & Y1 meets X;
defpred P10[set] means ex Y1,Y2 st Y1 in Y2 & Y2 in $1 & Y1 meets X;
defpred P9[set] means ex Y1,Y2,Y3 st Y1 in Y2 & Y2 in Y3 & Y3 in $1 & Y1
meets X;
defpred P8[set] means ex Y1,Y2,Y3,Y4 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4
in $1 & Y1 meets X;
defpred P7[set] means ex Y1,Y2,Y3,Y4,Y5 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 &
Y4 in Y5 & Y5 in $1 & Y1 meets X;
defpred P6[set] means ex Y1,Y2,Y3,Y4,Y5,Y6 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4
& Y4 in Y5 & Y5 in Y6 & Y6 in $1 & Y1 meets X;
defpred P5[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7 st Y1 in Y2 & Y2 in Y3 & Y3 in
Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in $1 & Y1 meets X;
defpred P4[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8 st Y1 in Y2 & Y2 in Y3 & Y3
in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in $1 & Y1 meets X;
defpred P3[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9 st Y1 in Y2 & Y2 in Y3 &
Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in $1 & Y1
meets X;
defpred P2[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA st Y1 in Y2 & Y2 in
Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA
& YA in $1 & Y1 meets X;
consider Z2 such that
A2: for Y holds Y in Z2 iff Y in union union X & P2[Y] from XBOOLE_0:sch
1;
consider Z7 such that
A3: for Y holds Y in Z7 iff Y in union union union union union union
union X & P7[Y] from XBOOLE_0:sch 1;
consider Z6 such that
A4: for Y holds Y in Z6 iff Y in union union union union union union X &
P6[Y] from XBOOLE_0:sch 1;
consider ZC such that
A5: for Y holds Y in ZC iff Y in union union union union union union
union union union union (union union X) & P12[Y] from XBOOLE_0:sch 1;
consider Z3 such that
A6: for Y holds Y in Z3 iff Y in union union union X & P3[Y] from
XBOOLE_0:sch 1;
consider Z5 such that
A7: for Y holds Y in Z5 iff Y in union union union union union X & P5[Y]
from XBOOLE_0:sch 1;
consider Z4 such that
A8: for Y holds Y in Z4 iff Y in union union union union X & P4[Y] from
XBOOLE_0:sch 1;
consider ZB such that
A9: for Y holds Y in ZB iff Y in union union union union union union
union union union union (union X) & P11[Y] from XBOOLE_0:sch 1;
consider ZA such that
A10: for Y holds Y in ZA iff Y in union union union union union union
union union union union X & P10[Y] from XBOOLE_0:sch 1;
consider Z9 such that
A11: for Y holds Y in Z9 iff Y in union union union union union union
union union union X & P9[Y] from XBOOLE_0:sch 1;
consider Z8 such that
A12: for Y holds Y in Z8 iff Y in union union union union union union
union union X & P8[Y] from XBOOLE_0:sch 1;
set V = (X \/ Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB \/ ZC;
A13: V = (X \/ (Z1 \/ Z2)) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB \/ ZC by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB
\/ ZC by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB
\/ ZC by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5) \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB
\/ ZC by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6) \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB
\/ ZC by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7) \/ Z8 \/ Z9 \/ ZA \/ ZB
\/ ZC by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8) \/ Z9 \/ ZA \/ ZB
\/ ZC by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9) \/ ZA \/ ZB
\/ ZC by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA) \/ ZB
\/ ZC by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB)
\/ ZC by XBOOLE_1:4
.= X \/ ((Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB \/ ZC) by XBOOLE_1:4;
assume X <> {};
then consider Y such that
A14: Y in V and
A15: Y misses V by MCART_1:1;
Y in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB or Y in ZC by A14,XBOOLE_0:def 3;
then
Y in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA or
Y in ZB or Y in ZC by XBOOLE_0:def 3;
then Y in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 or Y in
ZA or Y in ZB or Y in ZC by XBOOLE_0:def 3;
then Y in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 or Y in Z9 or
Y in ZA or Y in ZB or Y in ZC by XBOOLE_0:def 3;
then Y in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 or Y in Z8 or Y in
Z9 or Y in ZA or Y in ZB or Y in ZC by XBOOLE_0:def 3;
then
Y in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 or Y in Z7 or Y in Z8 or Y
in Z9 or Y in ZA or Y in ZB or Y in ZC by XBOOLE_0:def 3;
then
Y in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 or Y in Z6 or Y in Z7 or Y in Z8
or Y in Z9 or Y in ZA or Y in ZB or Y in ZC by XBOOLE_0:def 3;
then Y in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 or Y in Z5 or Y in Z6 or Y in Z7 or Y
in Z8 or Y in Z9 or Y in ZA or Y in ZB or Y in ZC by XBOOLE_0:def 3;
then Y in (X \/ Z1) \/ Z2 \/ Z3 or Y in Z4 or Y in Z5 or Y in Z6 or Y in Z7
or Y in Z8 or Y in Z9 or Y in ZA or Y in ZB or Y in ZC by XBOOLE_0:def 3;
then Y in (X \/ Z1) \/ Z2 or Y in Z3 or Y in Z4 or Y in Z5 or Y in Z6 or Y
in Z7 or Y in Z8 or Y in Z9 or Y in ZA or Y in ZB or Y in ZC by
XBOOLE_0:def 3;
then
A16: Y in X \/ Z1 or Y in Z2 or Y in Z3 or Y in Z4 or Y in Z5 or Y in Z6 or
Y in Z7 or Y in Z8 or Y in Z9 or Y in ZA or Y in ZB or Y in ZC by
XBOOLE_0:def 3;
assume
A17: not thesis;
per cases by A16,XBOOLE_0:def 3;
suppose
A18: Y in X;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB,YC such that
A19: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 &
Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC and
A20: YC in Y and
A21: not Y1 misses X by A17;
YC in union X by A18,A20,TARSKI:def 4;
then YC in Z1 by A1,A19,A21;
then YC in X \/ Z1 by XBOOLE_0:def 3;
then YC in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
then YC in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 3;
then YC in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then YC in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by A20,XBOOLE_0:3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_1:70;
then
Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_1:70;
then Y meets (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_1:70;
hence contradiction by A15,XBOOLE_1:70;
end;
suppose
A22: Y in Z1;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB such that
A23: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 &
Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB and
A24: YB in Y and
A25: Y1 meets X by A1;
Y in union X by A1,A22;
then YB in union union X by A24,TARSKI:def 4;
then YB in Z2 by A2,A23,A25;
then YB in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
then Y meets X \/ Z1 \/ Z2 by A24,XBOOLE_0:3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_1:70;
then
Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_1:70;
then Y meets (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_1:70;
hence contradiction by A15,XBOOLE_1:70;
end;
suppose
A26: Y in Z2;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA such that
A27: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 &
Y7 in Y8 & Y8 in Y9 & Y9 in YA and
A28: YA in Y and
A29: Y1 meets X by A2;
Y in union union X by A2,A26;
then YA in union union union X by A28,TARSKI:def 4;
then YA in Z3 by A6,A27,A29;
then YA in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then YA in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then YA in V by XBOOLE_0:def 3;
hence contradiction by A15,A28,XBOOLE_0:3;
end;
suppose
A30: Y in Z3;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9 such that
A31: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7
& Y7 in Y8 & Y8 in Y9 and
A32: Y9 in Y and
A33: Y1 meets X by A6;
Y in union union union X by A6,A30;
then Y9 in union union union union X by A32,TARSKI:def 4;
then Y9 in Z4 by A8,A31,A33;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y9 in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then Y9 in V by XBOOLE_0:def 3;
hence contradiction by A15,A32,XBOOLE_0:3;
end;
suppose
A34: Y in Z4;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8 such that
A35: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7
& Y7 in Y8 and
A36: Y8 in Y and
A37: Y1 meets X by A8;
Y in union union union union X by A8,A34;
then Y8 in union union union union union X by A36,TARSKI:def 4;
then Y8 in Z5 by A7,A35,A37;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y8 in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then Y8 in V by XBOOLE_0:def 3;
hence contradiction by A15,A36,XBOOLE_0:3;
end;
suppose
A38: Y in Z5;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7 such that
A39: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 and
A40: Y7 in Y and
A41: Y1 meets X by A7;
Y in union union union union union X by A7,A38;
then Y7 in union union union union union union X by A40,TARSKI:def 4;
then Y7 in Z6 by A4,A39,A41;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y7 in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then Y7 in V by XBOOLE_0:def 3;
hence contradiction by A15,A40,XBOOLE_0:3;
end;
suppose
A42: Y in Z6;
then consider Y1,Y2,Y3,Y4,Y5,Y6 such that
A43: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 and
A44: Y6 in Y and
A45: Y1 meets X by A4;
Y in union union union union union union X by A4,A42;
then Y6 in union union union union union union union X by A44,TARSKI:def 4;
then Y6 in Z7 by A3,A43,A45;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y6 in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then Y6 in V by XBOOLE_0:def 3;
hence contradiction by A15,A44,XBOOLE_0:3;
end;
suppose
A46: Y in Z7;
then consider Y1,Y2,Y3,Y4,Y5 such that
A47: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 and
A48: Y5 in Y and
A49: Y1 meets X by A3;
Y in union union union union union union union X by A3,A46;
then
Y5 in union union union union union union union union X by A48,TARSKI:def 4;
then Y5 in Z8 by A12,A47,A49;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y5 in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then Y5 in V by XBOOLE_0:def 3;
hence contradiction by A15,A48,XBOOLE_0:3;
end;
suppose
A50: Y in Z8;
then consider Y1,Y2,Y3,Y4 such that
A51: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 and
A52: Y4 in Y and
A53: Y1 meets X by A12;
Y in union union union union union union union union X by A12,A50;
then Y4 in union union union union union union union union union X by A52,
TARSKI:def 4;
then Y4 in Z9 by A11,A51,A53;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y4 in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then Y4 in V by XBOOLE_0:def 3;
hence contradiction by A15,A52,XBOOLE_0:3;
end;
suppose
A54: Y in Z9;
then consider Y1,Y2,Y3 such that
A55: Y1 in Y2 & Y2 in Y3 and
A56: Y3 in Y and
A57: Y1 meets X by A11;
Y in union union union union union union union union union X by A11,A54;
then Y3 in union union union union union union union union union union X
by A56,TARSKI:def 4;
then Y3 in ZA by A10,A55,A57;
then Y3 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y3 in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then Y3 in V by XBOOLE_0:def 3;
hence contradiction by A15,A56,XBOOLE_0:3;
end;
suppose
A58: Y in ZA;
then consider Y1,Y2 such that
A59: Y1 in Y2 and
A60: Y2 in Y and
A61: Y1 meets X by A10;
Y in union union union union union union union union union union X
by A10,A58;
then Y2 in union union union union union union union union union union (
union X) by A60,TARSKI:def 4;
then Y2 in ZB by A9,A59,A61;
then Y2 in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then Y2 in V by XBOOLE_0:def 3;
hence contradiction by A15,A60,XBOOLE_0:3;
end;
suppose
A62: Y in ZB;
then consider Y1 such that
A63: Y1 in Y and
A64: Y1 meets X by A9;
Y in union union union union union union union union union union (
union X) by A9,A62;
then Y1 in union union union union union union union union union union (
union union X) by A63,TARSKI:def 4;
then Y1 in ZC by A5,A64;
then Y1 in V by XBOOLE_0:def 3;
hence contradiction by A15,A63,XBOOLE_0:3;
end;
suppose
Y in ZC;
then Y meets X by A5;
hence contradiction by A13,A15,XBOOLE_1:70;
end;
end;
theorem Th117:
X <> {} implies ex Y st Y in X & for Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9
,YA,YB,YC,YD st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7
& Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC & YC in YD & YD in Y
holds Y1 misses X
proof
defpred P1[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB,YC st Y1 in Y2 &
Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9
in YA & YA in YB & YB in YC & YC in $1 & Y1 meets X;
consider Z1 such that
A1: for Y holds Y in Z1 iff Y in union X & P1[Y] from XBOOLE_0:sch 1;
defpred P13[set] means $1 meets X;
defpred P12[set] means ex Y1 st Y1 in $1 & Y1 meets X;
defpred P11[set] means ex Y1,Y2 st Y1 in Y2 & Y2 in $1 & Y1 meets X;
defpred P10[set] means ex Y1,Y2,Y3 st Y1 in Y2 & Y2 in Y3 & Y3 in $1 & Y1
meets X;
defpred P9[set] means ex Y1,Y2,Y3,Y4 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4
in $1 & Y1 meets X;
defpred P8[set] means ex Y1,Y2,Y3,Y4,Y5 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 &
Y4 in Y5 & Y5 in $1 & Y1 meets X;
defpred P7[set] means ex Y1,Y2,Y3,Y4,Y5,Y6 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4
& Y4 in Y5 & Y5 in Y6 & Y6 in $1 & Y1 meets X;
defpred P6[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7 st Y1 in Y2 & Y2 in Y3 & Y3 in
Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in $1 & Y1 meets X;
defpred P5[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8 st Y1 in Y2 & Y2 in Y3 & Y3
in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in $1 & Y1 meets X;
defpred P4[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9 st Y1 in Y2 & Y2 in Y3 &
Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in $1 & Y1
meets X;
defpred P3[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA st Y1 in Y2 & Y2 in
Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA
& YA in $1 & Y1 meets X;
defpred P2[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB st Y1 in Y2 & Y2
in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in
YA & YA in YB & YB in $1 & Y1 meets X;
consider Z2 such that
A2: for Y holds Y in Z2 iff Y in union union X & P2[Y] from XBOOLE_0:sch
1;
consider Z9 such that
A3: for Y holds Y in Z9 iff Y in union union union union union union
union union union X & P9[Y] from XBOOLE_0:sch 1;
consider Z8 such that
A4: for Y holds Y in Z8 iff Y in union union union union union union
union union X & P8[Y] from XBOOLE_0:sch 1;
consider ZD such that
A5: for Y holds Y in ZD iff Y in union union union union union union
union union union union (union union union X) & P13[Y] from XBOOLE_0:sch 1;
consider Z3 such that
A6: for Y holds Y in Z3 iff Y in union union union X & P3[Y] from
XBOOLE_0:sch 1;
consider Z7 such that
A7: for Y holds Y in Z7 iff Y in union union union union union union
union X & P7[Y] from XBOOLE_0:sch 1;
consider Z6 such that
A8: for Y holds Y in Z6 iff Y in union union union union union union X &
P6[Y] from XBOOLE_0:sch 1;
consider ZB such that
A9: for Y holds Y in ZB iff Y in union union union union union union
union union union union (union X) & P11[Y] from XBOOLE_0:sch 1;
consider ZA such that
A10: for Y holds Y in ZA iff Y in union union union union union union
union union union union X & P10[Y] from XBOOLE_0:sch 1;
consider ZC such that
A11: for Y holds Y in ZC iff Y in union union union union union union
union union union union (union union X) & P12[Y] from XBOOLE_0:sch 1;
consider Z5 such that
A12: for Y holds Y in Z5 iff Y in union union union union union X & P5[Y]
from XBOOLE_0:sch 1;
consider Z4 such that
A13: for Y holds Y in Z4 iff Y in union union union union X & P4[Y] from
XBOOLE_0:sch 1;
set V = (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB \/ ZC \/ ZD;
A14: V = (X \/ (Z1 \/ Z2) \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB \/ ZC \/ ZD by XBOOLE_1:4
.= (X \/ (Z1 \/ Z2 \/ Z3)) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB \/ ZC \/ ZD by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB
\/ ZC \/ ZD by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5) \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB
\/ ZC \/ ZD by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6) \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB
\/ ZC \/ ZD by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7) \/ Z8 \/ Z9 \/ ZA \/ ZB
\/ ZC \/ ZD by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8) \/ Z9 \/ ZA \/ ZB
\/ ZC \/ ZD by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9) \/ ZA \/ ZB
\/ ZC \/ ZD by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA) \/ ZB
\/ ZC \/ ZD by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB)
\/ ZC \/ ZD by XBOOLE_1:4
.= X \/ ((Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB \/ ZC) \/ ZD by XBOOLE_1:4
.= X \/ ((Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB \/ ZC \/ ZD) by XBOOLE_1:4;
assume X <> {};
then consider Y such that
A15: Y in V and
A16: Y misses V by MCART_1:1;
Y in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB \/ ZC or Y in ZD by A15,XBOOLE_0:def 3;
then
Y in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB or Y in ZC or Y in ZD by XBOOLE_0:def 3;
then
Y in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA or
Y in ZB or Y in ZC or Y in ZD by XBOOLE_0:def 3;
then Y in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 or Y in
ZA or Y in ZB or Y in ZC or Y in ZD by XBOOLE_0:def 3;
then Y in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 or Y in Z9 or
Y in ZA or Y in ZB or Y in ZC or Y in ZD by XBOOLE_0:def 3;
then Y in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 or Y in Z8 or Y in
Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD by XBOOLE_0:def 3;
then
Y in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 or Y in Z7 or Y in Z8 or Y
in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD by XBOOLE_0:def 3;
then
Y in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 or Y in Z6 or Y in Z7 or Y in Z8
or Y in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD by XBOOLE_0:def 3;
then Y in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 or Y in Z5 or Y in Z6 or Y in Z7 or Y
in Z8 or Y in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD by
XBOOLE_0:def 3;
then Y in (X \/ Z1 \/ Z2 \/ Z3) or Y in Z4 or Y in Z5 or Y in Z6 or Y in Z7
or Y in Z8 or Y in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD by
XBOOLE_0:def 3;
then Y in (X \/ Z1) \/ Z2 or Y in Z3 or Y in Z4 or Y in Z5 or Y in Z6 or Y
in Z7 or Y in Z8 or Y in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD by
XBOOLE_0:def 3;
then
A17: Y in X \/ Z1 or Y in Z2 or Y in Z3 or Y in Z4 or Y in Z5 or Y in Z6 or
Y in Z7 or Y in Z8 or Y in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD by
XBOOLE_0:def 3;
assume
A18: not thesis;
per cases by A17,XBOOLE_0:def 3;
suppose
A19: Y in X;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB,YC,YD such that
A20: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 &
Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC & YC in YD and
A21: YD in Y and
A22: not Y1 misses X by A18;
YD in union X by A19,A21,TARSKI:def 4;
then YD in Z1 by A1,A20,A22;
then YD in X \/ Z1 by XBOOLE_0:def 3;
then YD in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
then YD in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 3;
then YD in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then YD in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by A21,XBOOLE_0:3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_1:70;
then
Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_1:70;
then Y meets (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_1:70;
then Y meets (X \/ Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_1:70;
hence contradiction by A16,XBOOLE_1:70;
end;
suppose
A23: Y in Z1;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB,YC such that
A24: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 &
Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC and
A25: YC in Y and
A26: Y1 meets X by A1;
Y in union X by A1,A23;
then YC in union union X by A25,TARSKI:def 4;
then YC in Z2 by A2,A24,A26;
then YC in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
then Y meets X \/ Z1 \/ Z2 by A25,XBOOLE_0:3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_1:70;
then
Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_1:70;
then Y meets (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_1:70;
then Y meets (X \/ Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_1:70;
hence contradiction by A16,XBOOLE_1:70;
end;
suppose
A27: Y in Z2;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB such that
A28: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 &
Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB and
A29: YB in Y and
A30: Y1 meets X by A2;
Y in union union X by A2,A27;
then YB in union union union X by A29,TARSKI:def 4;
then YB in Z3 by A6,A28,A30;
then YB in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then YB in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then YB in (X \/ Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_0:def 3;
then YB in V by XBOOLE_0:def 3;
hence contradiction by A16,A29,XBOOLE_0:3;
end;
suppose
A31: Y in Z3;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA such that
A32: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7
& Y7 in Y8 & Y8 in Y9 & Y9 in YA and
A33: YA in Y and
A34: Y1 meets X by A6;
Y in union union union X by A6,A31;
then YA in union union union union X by A33,TARSKI:def 4;
then YA in Z4 by A13,A32,A34;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then YA in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then YA in (X \/ Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_0:def 3;
then YA in V by XBOOLE_0:def 3;
hence contradiction by A16,A33,XBOOLE_0:3;
end;
suppose
A35: Y in Z4;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9 such that
A36: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7
& Y7 in Y8 & Y8 in Y9 and
A37: Y9 in Y and
A38: Y1 meets X by A13;
Y in union union union union X by A13,A35;
then Y9 in union union union union union X by A37,TARSKI:def 4;
then Y9 in Z5 by A12,A36,A38;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y9 in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then Y9 in (X \/ Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_0:def 3;
then Y9 in V by XBOOLE_0:def 3;
hence contradiction by A16,A37,XBOOLE_0:3;
end;
suppose
A39: Y in Z5;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8 such that
A40: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7
& Y7 in Y8 and
A41: Y8 in Y and
A42: Y1 meets X by A12;
Y in union union union union union X by A12,A39;
then Y8 in union union union union union union X by A41,TARSKI:def 4;
then Y8 in Z6 by A8,A40,A42;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y8 in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then Y8 in (X \/ Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_0:def 3;
then Y8 in V by XBOOLE_0:def 3;
hence contradiction by A16,A41,XBOOLE_0:3;
end;
suppose
A43: Y in Z6;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7 such that
A44: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 and
A45: Y7 in Y and
A46: Y1 meets X by A8;
Y in union union union union union union X by A8,A43;
then Y7 in union union union union union union union X by A45,TARSKI:def 4;
then Y7 in Z7 by A7,A44,A46;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y7 in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then Y7 in (X \/ Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_0:def 3;
then Y7 in V by XBOOLE_0:def 3;
hence contradiction by A16,A45,XBOOLE_0:3;
end;
suppose
A47: Y in Z7;
then consider Y1,Y2,Y3,Y4,Y5,Y6 such that
A48: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 and
A49: Y6 in Y and
A50: Y1 meets X by A7;
Y in union union union union union union union X by A7,A47;
then
Y6 in union union union union union union union union X by A49,TARSKI:def 4;
then Y6 in Z8 by A4,A48,A50;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y6 in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then Y6 in (X \/ Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_0:def 3;
then Y6 in V by XBOOLE_0:def 3;
hence contradiction by A16,A49,XBOOLE_0:3;
end;
suppose
A51: Y in Z8;
then consider Y1,Y2,Y3,Y4,Y5 such that
A52: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 and
A53: Y5 in Y and
A54: Y1 meets X by A4;
Y in union union union union union union union union X by A4,A51;
then Y5 in union union union union union union union union union X by A53,
TARSKI:def 4;
then Y5 in Z9 by A3,A52,A54;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y5 in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then Y5 in ((X \/ Z1) \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_0:def 3;
then Y5 in V by XBOOLE_0:def 3;
hence contradiction by A16,A53,XBOOLE_0:3;
end;
suppose
A55: Y in Z9;
then consider Y1,Y2,Y3,Y4 such that
A56: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 and
A57: Y4 in Y and
A58: Y1 meets X by A3;
Y in union union union union union union union union union X by A3,A55;
then Y4 in union union union union union union union union union union X
by A57,TARSKI:def 4;
then Y4 in ZA by A10,A56,A58;
then Y4 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y4 in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then Y4 in (X \/ Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_0:def 3;
then Y4 in V by XBOOLE_0:def 3;
hence contradiction by A16,A57,XBOOLE_0:3;
end;
suppose
A59: Y in ZA;
then consider Y1,Y2,Y3 such that
A60: Y1 in Y2 & Y2 in Y3 and
A61: Y3 in Y and
A62: Y1 meets X by A10;
Y in union union union union union union union union union union X
by A10,A59;
then Y3 in union union union union union union union union union union (
union X) by A61,TARSKI:def 4;
then Y3 in ZB by A9,A60,A62;
then Y3 in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then Y3 in (X \/ Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_0:def 3;
then Y3 in V by XBOOLE_0:def 3;
hence contradiction by A16,A61,XBOOLE_0:3;
end;
suppose
A63: Y in ZB;
then consider Y1,Y2 such that
A64: Y1 in Y2 and
A65: Y2 in Y and
A66: Y1 meets X by A9;
Y in union union union union union union union union union union (
union X) by A9,A63;
then Y2 in union union union union union union union union union union (
union union X) by A65,TARSKI:def 4;
then Y2 in ZC by A11,A64,A66;
then Y2 in ((X \/ Z1) \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_0:def 3;
then Y2 in V by XBOOLE_0:def 3;
hence contradiction by A16,A65,XBOOLE_0:3;
end;
suppose
A67: Y in ZC;
then consider Y1 such that
A68: Y1 in Y and
A69: Y1 meets X by A11;
Y in union union union union union union union union union union (
union union X) by A11,A67;
then Y1 in union union union union union union union union union union (
union union union X) by A68,TARSKI:def 4;
then Y1 in ZD by A5,A69;
then Y1 in V by XBOOLE_0:def 3;
hence contradiction by A16,A68,XBOOLE_0:3;
end;
suppose
Y in ZD;
then Y meets X by A5;
hence contradiction by A14,A16,XBOOLE_1:70;
end;
end;
::
:: Tuples for n=8
::
definition
let x1,x2,x3,x4,x5,x6,x7,x8;
func [x1,x2,x3,x4,x5,x6,x7,x8] equals
[[x1,x2,x3,x4,x5,x6,x7],x8];
correctness;
end;
theorem
[x1,x2,x3,x4,x5,x6,x7,x8] = [[[[[[[x1,x2],x3],x4],x5],x6],x7],x8]
proof
thus [x1,x2,x3,x4,x5,x6,x7,x8] = [[[[[[x1,x2,x3],x4],x5],x6],x7],x8] by
MCART_1:def 4
.= [[[[[[[x1,x2],x3],x4],x5],x6],x7],x8] by MCART_1:def 3;
end;
theorem
[x1,x2,x3,x4,x5,x6,x7,x8] = [[x1,x2,x3,x4,x5,x6],x7,x8] by MCART_1:def 3;
theorem
[x1,x2,x3,x4,x5,x6,x7,x8] = [[x1,x2,x3,x4,x5],x6,x7,x8] by MCART_1:27;
theorem
[x1,x2,x3,x4,x5,x6,x7,x8] = [[x1,x2,x3,x4],x5,x6,x7,x8] by Th3;
theorem
[x1,x2,x3,x4,x5,x6,x7,x8] = [[x1,x2,x3],x4,x5,x6,x7,x8]
proof
thus [x1,x2,x3,x4,x5,x6,x7,x8] = [[[[[[[x1,x2],x3],x4],x5],x6],x7],x8] by Th3
.= [[[x1,x2],x3],x4,x5,x6,x7,x8] by Th41
.= [[x1,x2,x3],x4,x5,x6,x7,x8] by MCART_1:def 3;
end;
theorem
[x1,x2,x3,x4,x5,x6,x7,x8] = [[x1,x2],x3,x4,x5,x6,x7,x8] by Th6;
theorem Th124:
[x1,x2,x3,x4,x5,x6,x7,x8] = [y1,y2,y3,y4,y5,y6,y7,y8] implies
x1 = y1 & x2 = y2 & x3 = y3 & x4 = y4 & x5 = y5 & x6 = y6 & x7 = y7 & x8 = y8
proof
assume
A1: [x1,x2,x3,x4,x5,x6,x7,x8] = [y1,y2,y3,y4,y5,y6,y7,y8];
then [x1,x2,x3,x4,x5,x6,x7] = [y1,y2,y3,y4,y5,y6,y7] by ZFMISC_1:27;
hence thesis by A1,Th83,ZFMISC_1:27;
end;
theorem Th125:
X <> {} implies ex y st y in X & not ex x1,x2,x3,x4,x5,x6,x7,x8
st (x1 in X or x2 in X) & y = [x1,x2,x3,x4,x5,x6,x7,x8]
proof
assume X <> {};
then consider Y such that
A1: Y in X and
A2: for Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB,YC,YD st Y1 in Y2 & Y2 in Y3 &
Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA
in YB & YB in YC & YC in YD & YD in Y holds Y1 misses X by Th117;
take y = Y;
thus y in X by A1;
given x1,x2,x3,x4,x5,x6,x7,x8 such that
A3: x1 in X or x2 in X and
A4: y = [x1,x2,x3,x4,x5,x6,x7,x8];
set Y1 = { x1, x2 }, Y2 = { Y1, {x1} }, Y3 = { Y2, x3 }, Y4 = { Y3, {Y2} },
Y5 = { Y4, x4 }, Y6 = { Y5, {Y4} }, Y7 = { Y6, x5 }, Y8 = { Y7, {Y6} }, Y9 = {
Y8, x6 }, YA = { Y9, {Y8} }, YB = { YA, x7 }, YC = { YB, {YA} }, YD = { YC, x8
};
A5: Y3 in Y4 & Y4 in Y5 by TARSKI:def 2;
Y = [[[[[[[x1,x2],x3],x4],x5],x6],x7],x8] by A4,Th3
.= [[[[[[ Y2,x3],x4],x5],x6],x7],x8 ] by TARSKI:def 5
.= [[[[[ Y4,x4],x5],x6],x7],x8 ] by TARSKI:def 5
.= [[[[ Y6,x5 ],x6],x7],x8 ] by TARSKI:def 5
.= [[[ Y8,x6],x7],x8 ] by TARSKI:def 5
.= [[ YA,x7],x8 ] by TARSKI:def 5
.= [ YC,x8 ] by TARSKI:def 5
.= { YD, { YC } } by TARSKI:def 5;
then
A6: YD in Y by TARSKI:def 2;
A7: Y5 in Y6 & Y6 in Y7 by TARSKI:def 2;
A8: YB in YC & YC in YD by TARSKI:def 2;
A9: Y9 in YA & YA in YB by TARSKI:def 2;
x1 in Y1 & x2 in Y1 by TARSKI:def 2;
then
A10: not Y1 misses X by A3,XBOOLE_0:3;
A11: Y7 in Y8 & Y8 in Y9 by TARSKI:def 2;
Y1 in Y2 & Y2 in Y3 by TARSKI:def 2;
hence contradiction by A2,A10,A5,A7,A11,A9,A8,A6;
end;
::
:: Cartesian products of eight sets
::
definition
let X1,X2,X3,X4,X5,X6,X7,X8;
func [:X1,X2,X3,X4,X5,X6,X7,X8:] -> set equals
[:[: X1,X2,X3,X4,X5,X6,X7 :],
X8 :];
correctness;
end;
theorem Th126:
[:X1,X2,X3,X4,X5,X6,X7,X8:] = [:[:[:[:[:[:[:X1,X2:],X3:],X4:],
X5:],X6:],X7:],X8:]
proof
thus [:X1,X2,X3,X4,X5,X6,X7,X8:] = [:[:[:[:[:[:X1,X2,X3:],X4:],X5:],X6:],X7
:],X8:] by ZFMISC_1:def 4
.= [:[:[:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:],X7:],X8:] by ZFMISC_1:def 3;
end;
theorem
[:X1,X2,X3,X4,X5,X6,X7,X8:] = [:[:X1,X2,X3,X4,X5,X6:],X7,X8:] by
ZFMISC_1:def 3;
theorem
[:X1,X2,X3,X4,X5,X6,X7,X8:] = [:[:X1,X2,X3,X4,X5:],X6,X7,X8:] by MCART_1:49;
theorem
[:X1,X2,X3,X4,X5,X6,X7,X8:] = [:[:X1,X2,X3,X4:],X5,X6,X7,X8:] by Th9;
theorem
[:X1,X2,X3,X4,X5,X6,X7,X8:] = [:[:X1,X2,X3:],X4,X5,X6,X7,X8:]
proof
thus [:X1,X2,X3,X4,X5,X6,X7,X8:] = [:[:[:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:]
,X7:],X8:] by Th126
.= [:[:[:X1,X2:],X3:],X4,X5,X6,X7,X8:] by Th47
.= [:[:X1,X2,X3:],X4,X5,X6,X7,X8:] by ZFMISC_1:def 3;
end;
theorem
[:X1,X2,X3,X4,X5,X6,X7,X8:] = [:[:X1,X2:],X3,X4,X5,X6,X7,X8:] by Th12;
theorem Th132:
X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {}
& X7 <> {} & X8 <> {} iff [:X1,X2,X3,X4,X5,X6,X7,X8:] <> {}
proof
X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {} & X7 <>
{} iff [:X1,X2,X3,X4,X5,X6,X7:] <> {} by Th90;
hence thesis by ZFMISC_1:90;
end;
theorem Th133:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} &
X8<>{} implies ( [:X1,X2,X3,X4,X5,X6,X7,X8:] = [:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8:]
implies X1=Y1 & X2=Y2 & X3=Y3 & X4=Y4 & X5=Y5 & X6=Y6 & X7=Y7 & X8=Y8)
proof
assume
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{};
then
A2: [:X1,X2,X3,X4,X5,X6,X7:] <> {} by Th90;
assume
A3: X8<>{};
assume
A4: [:X1,X2,X3,X4,X5,X6,X7,X8:] = [:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8:];
then [:X1,X2,X3,X4,X5,X6,X7:] = [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:] by A2,A3,
ZFMISC_1:110;
hence thesis by A1,A2,A3,A4,Th91,ZFMISC_1:110;
end;
theorem
[:X1,X2,X3,X4,X5,X6,X7,X8:]<>{} & [:X1,X2,X3,X4,X5,X6,X7,X8:] = [:Y1,
Y2,Y3,Y4,Y5,Y6,Y7,Y8:] implies X1=Y1 & X2=Y2 & X3=Y3 & X4=Y4 & X5=Y5 & X6=Y6 &
X7=Y7 & X8=Y8
proof
assume
A1: [:X1,X2,X3,X4,X5,X6,X7,X8:]<>{};
then
A2: X3<>{} & X4<>{} by Th132;
A3: X7<>{} & X8<> {} by A1,Th132;
A4: X5<>{} & X6<>{} by A1,Th132;
X1<>{} & X2<>{} by A1,Th132;
hence thesis by A2,A4,A3,Th133;
end;
theorem
[:X,X,X,X,X,X,X,X:] = [:Y,Y,Y,Y,Y,Y,Y,Y:] implies X = Y
proof
assume [:X,X,X,X,X,X,X,X:] = [:Y,Y,Y,Y,Y,Y,Y,Y:];
then X<>{} or Y<>{} implies thesis by Th133;
hence thesis;
end;
theorem Th136:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} &
X8<>{} implies for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:] ex xx1,xx2,
xx3,xx4,xx5,xx6,xx7,xx8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8]
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} and
A2: X8<>{};
let x be Element of [:X1,X2,X3,X4,X5,X6,X7,X8:];
reconsider x9=x as Element of [:[:X1,X2,X3,X4,X5,X6,X7:],X8:];
[:X1,X2,X3,X4,X5,X6,X7:] <> {} by A1,Th90;
then consider
x1234567 being (Element of [:X1,X2,X3,X4,X5,X6,X7:]), xx8 such that
A3: x9 = [x1234567,xx8] by A2,Lm1;
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7 such that
A4: x1234567 = [xx1,xx2,xx3,xx4,xx5,xx6,xx7] by A1,Th94;
take xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8;
thus thesis by A3,A4;
end;
definition
let X1,X2,X3,X4,X5,X6,X7,X8;
assume
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <> {};
let x be Element of [:X1,X2,X3,X4,X5,X6,X7,X8:];
func x`1 -> Element of X1 means
:Def27:
x = [x1,x2,x3,x4,x5,x6,x7,x8] implies it = x1;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 such that
A2: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] by A1,Th136;
take xx1;
thus thesis by A2,Th124;
end;
uniqueness
proof
let y,z be Element of X1;
assume
A3: x = [x1,x2,x3,x4,x5,x6,x7,x8] implies y = x1;
assume
A4: x = [x1,x2,x3,x4,x5,x6,x7,x8] implies z = x1;
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 such that
A5: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] by A1,Th136;
y = xx1 by A5,A3;
hence thesis by A5,A4;
end;
func x`2 -> Element of X2 means
:Def28:
x = [x1,x2,x3,x4,x5,x6,x7,x8] implies it = x2;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 such that
A6: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] by A1,Th136;
take xx2;
thus thesis by A6,Th124;
end;
uniqueness
proof
let y,z be Element of X2;
assume
A7: x = [x1,x2,x3,x4,x5,x6,x7,x8] implies y = x2;
assume
A8: x = [x1,x2,x3,x4,x5,x6,x7,x8] implies z = x2;
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 such that
A9: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] by A1,Th136;
y = xx2 by A9,A7;
hence thesis by A9,A8;
end;
func x`3 -> Element of X3 means
:Def29:
x = [x1,x2,x3,x4,x5,x6,x7,x8] implies it = x3;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 such that
A10: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] by A1,Th136;
take xx3;
thus thesis by A10,Th124;
end;
uniqueness
proof
let y,z be Element of X3;
assume
A11: x = [x1,x2,x3,x4,x5,x6,x7,x8] implies y = x3;
assume
A12: x = [x1,x2,x3,x4,x5,x6,x7,x8] implies z = x3;
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 such that
A13: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] by A1,Th136;
y = xx3 by A13,A11;
hence thesis by A13,A12;
end;
func x`4 -> Element of X4 means
:Def30:
x = [x1,x2,x3,x4,x5,x6,x7,x8] implies it = x4;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 such that
A14: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] by A1,Th136;
take xx4;
thus thesis by A14,Th124;
end;
uniqueness
proof
let y,z be Element of X4;
assume
A15: x = [x1,x2,x3,x4,x5,x6,x7,x8] implies y = x4;
assume
A16: x = [x1,x2,x3,x4,x5,x6,x7,x8] implies z = x4;
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 such that
A17: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] by A1,Th136;
y = xx4 by A17,A15;
hence thesis by A17,A16;
end;
func x`5 -> Element of X5 means
:Def31:
x = [x1,x2,x3,x4,x5,x6,x7,x8] implies it = x5;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 such that
A18: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] by A1,Th136;
take xx5;
thus thesis by A18,Th124;
end;
uniqueness
proof
let y,z be Element of X5;
assume
A19: x = [x1,x2,x3,x4,x5,x6,x7,x8] implies y = x5;
assume
A20: x = [x1,x2,x3,x4,x5,x6,x7,x8] implies z = x5;
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 such that
A21: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] by A1,Th136;
y = xx5 by A21,A19;
hence thesis by A21,A20;
end;
func x`6 -> Element of X6 means
:Def32:
x = [x1,x2,x3,x4,x5,x6,x7,x8] implies it = x6;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 such that
A22: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] by A1,Th136;
take xx6;
thus thesis by A22,Th124;
end;
uniqueness
proof
let y,z be Element of X6;
assume
A23: x = [x1,x2,x3,x4,x5,x6,x7,x8] implies y = x6;
assume
A24: x = [x1,x2,x3,x4,x5,x6,x7,x8] implies z = x6;
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 such that
A25: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] by A1,Th136;
y = xx6 by A25,A23;
hence thesis by A25,A24;
end;
func x`7 -> Element of X7 means
:Def33:
x = [x1,x2,x3,x4,x5,x6,x7,x8] implies it = x7;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 such that
A26: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] by A1,Th136;
take xx7;
thus thesis by A26,Th124;
end;
uniqueness
proof
let y,z be Element of X7;
assume
A27: x = [x1,x2,x3,x4,x5,x6,x7,x8] implies y = x7;
assume
A28: x = [x1,x2,x3,x4,x5,x6,x7,x8] implies z = x7;
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 such that
A29: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] by A1,Th136;
y = xx7 by A29,A27;
hence thesis by A29,A28;
end;
func x`8 -> Element of X8 means
:Def34:
x = [x1,x2,x3,x4,x5,x6,x7,x8] implies it = x8;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 such that
A30: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] by A1,Th136;
take xx8;
thus thesis by A30,Th124;
end;
uniqueness
proof
let y,z be Element of X8;
assume
A31: x = [x1,x2,x3,x4,x5,x6,x7,x8] implies y = x8;
assume
A32: x = [x1,x2,x3,x4,x5,x6,x7,x8] implies z = x8;
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 such that
A33: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] by A1,Th136;
y = xx8 by A33,A31;
hence thesis by A33,A32;
end;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8<>{}
implies for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:] for x1,x2,x3,x4,x5,
x6,x7,x8 st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds x`1 = x1 & x`2 = x2 & x`3 = x3
& x`4 = x4 & x`5 = x5 & x`6 = x6 & x`7 = x7 & x`8 =x8 by Def27,Def28,Def29
,Def30,Def31,Def32,Def33,Def34;
theorem Th138:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} &
X8<>{} implies for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:] holds x = [x
`1,x`2,x`3,x`4,x`5,x`6,x`7,x`8]
proof
assume
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <> {};
let x be Element of [:X1,X2,X3,X4,X5,X6,X7,X8:];
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 such that
A2: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] by A1,Th136;
thus x = [x`1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] by A1,A2,Def27
.= [x`1,x`2,xx3,xx4,xx5,xx6,xx7,xx8] by A1,A2,Def28
.= [x`1,x`2,x`3,xx4,xx5,xx6,xx7,xx8] by A1,A2,Def29
.= [x`1,x`2,x`3,x`4,xx5,xx6,xx7,xx8] by A1,A2,Def30
.= [x`1,x`2,x`3,x`4,x`5,xx6,xx7,xx8] by A1,A2,Def31
.= [x`1,x`2,x`3,x`4,x`5,x`6,xx7,xx8] by A1,A2,Def32
.= [x`1,x`2,x`3,x`4,x`5,x`6,x`7,xx8] by A1,A2,Def33
.= [x`1,x`2,x`3,x`4,x`5,x`6,x`7,x`8] by A1,A2,Def34;
end;
theorem Th139:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} &
X8<>{} implies for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:] holds x`1 = (
x qua set)`1`1`1`1`1`1`1 & x`2 = (x qua set)`1`1`1`1`1`1`2 & x`3 = (x qua set)
`1`1`1`1`1`2 & x`4 = (x qua set)`1`1`1`1`2 & x`5 = (x qua set)`1`1`1`2 & x`6 =
(x qua set)`1`1`2 & x`7 = (x qua set)`1`2 & x`8 = (x qua set)`2
proof
assume
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <> {};
let x be Element of [:X1,X2,X3,X4,X5,X6,X7,X8:];
thus x`1 = [ x`1, x`2]`1 by MCART_1:7
.= [[x`1, x`2],x`3]`1`1 by MCART_1:7
.= [ x`1, x`2 ,x`3]`1`1 by MCART_1:def 3
.= [[x`1, x`2 ,x`3],x`4]`1`1`1 by MCART_1:7
.= [ x`1, x`2 ,x`3 ,x`4]`1`1`1 by MCART_1:def 4
.= [[x`1, x`2 ,x`3 ,x`4], x`5]`1`1`1`1 by MCART_1:7
.= [[x`1, x`2 ,x`3 ,x`4, x`5],x`6]`1`1`1`1`1 by MCART_1:7
.= [[x`1, x`2 ,x`3 ,x`4, x`5, x`6],x`7]`1`1`1`1`1`1 by MCART_1:7
.= [ x`1, x`2 ,x`3 ,x`4 ,x`5, x`6, x`7, x`8]`1`1`1`1`1`1`1 by MCART_1:7
.= (x qua set)`1`1`1`1`1`1`1 by A1,Th138;
thus x`2 = [ x`1, x`2]`2 by MCART_1:7
.= [[x`1, x`2],x`3]`1`2 by MCART_1:7
.= [ x`1, x`2, x`3]`1`2 by MCART_1:def 3
.= [[x`1, x`2, x`3],x`4]`1`1`2 by MCART_1:7
.= [ x`1, x`2, x`3, x`4]`1`1`2 by MCART_1:def 4
.= [[x`1, x`2, x`3, x`4 ], x`5]`1`1`1`2 by MCART_1:7
.= [[x`1, x`2, x`3, x`4, x`5],x`6]`1`1`1`1`2 by MCART_1:7
.= [[x`1, x`2, x`3, x`4, x`5, x`6],x`7]`1`1`1`1`1`2 by MCART_1:7
.= [ x`1, x`2, x`3, x`4, x`5, x`6, x`7, x`8]`1`1`1`1`1`1`2 by MCART_1:7
.= (x qua set)`1`1`1`1`1`1`2 by A1,Th138;
thus x`3 = [[x`1, x`2],x`3]`2 by MCART_1:7
.= [ x`1, x`2, x`3]`2 by MCART_1:def 3
.= [[x`1, x`2, x`3],x`4]`1`2 by MCART_1:7
.= [ x`1, x`2, x`3, x`4]`1`2 by MCART_1:def 4
.= [[x`1, x`2, x`3, x`4],x`5]`1`1`2 by MCART_1:7
.= [[x`1, x`2, x`3, x`4, x`5],x`6]`1`1`1`2 by MCART_1:7
.= [[x`1, x`2, x`3, x`4, x`5, x`6],x`7]`1`1`1`1`2 by MCART_1:7
.= [ x`1, x`2, x`3, x`4, x`5, x`6, x`7, x`8]`1`1`1`1`1`2 by MCART_1:7
.= (x qua set)`1`1`1`1`1`2 by A1,Th138;
thus x`4 = [[x`1,x`2,x`3],x`4]`2 by MCART_1:7
.= [ x`1,x`2,x`3, x`4]`2 by MCART_1:def 4
.= [[x`1,x`2,x`3, x`4],x`5]`1`2 by MCART_1:7
.= [[x`1,x`2,x`3, x`4, x`5],x`6]`1`1`2 by MCART_1:7
.= [[x`1,x`2,x`3, x`4, x`5, x`6],x`7]`1`1`1`2 by MCART_1:7
.= [ x`1,x`2,x`3, x`4, x`5, x`6, x`7, x`8]`1`1`1`1`2 by MCART_1:7
.= (x qua set)`1`1`1`1`2 by A1,Th138;
thus x`5 = [[x`1,x`2,x`3,x`4],x`5]`2 by MCART_1:7
.= [[x`1,x`2,x`3,x`4,x`5],x`6]`1`2 by MCART_1:7
.= [[x`1,x`2,x`3,x`4,x`5,x`6],x`7]`1`1`2 by MCART_1:7
.= [ x`1,x`2,x`3,x`4,x`5, x`6,x`7,x`8]`1`1`1`2 by MCART_1:7
.= (x qua set)`1`1`1`2 by A1,Th138;
thus x`6 = [[x`1,x`2,x`3,x`4,x`5],x`6]`2 by MCART_1:7
.= [[x`1,x`2,x`3,x`4,x`5,x`6],x`7]`1`2 by MCART_1:7
.= [ x`1,x`2,x`3,x`4,x`5,x`6,x`7, x`8]`1`1`2 by MCART_1:7
.= (x qua set)`1`1`2 by A1,Th138;
thus x`7 = [[x`1,x`2,x`3,x`4,x`5,x`6],x`7]`2 by MCART_1:7
.= [ x`1,x`2,x`3,x`4,x`5,x`6,x`7, x`8]`1`2 by MCART_1:7
.= (x qua set)`1`2 by A1,Th138;
thus x`8 = [ x`1,x`2,x`3,x`4,x`5,x`6,x`7, x`8]`2 by MCART_1:7
.= (x qua set)`2 by A1,Th138;
end;
theorem
X1 c= [:X1,X2,X3,X4,X5,X6,X7,X8:] or X1 c= [:X2,X3,X4,X5,X6,X7,X8,X1:]
or X1 c= [:X3,X4,X5,X6,X7,X8,X1,X2:] or X1 c= [:X4,X5,X6,X7,X8,X1,X2,X3:] or X1
c= [:X5,X6,X7,X8,X1,X2,X3,X4:] or X1 c= [:X6,X7,X8,X1,X2,X3,X4,X5:] or X1 c= [:
X7,X8,X1,X2,X3,X4,X5,X6:] or X1 c= [:X8,X1,X2,X3,X4,X5,X6,X7:] implies X1 = {}
proof
assume that
A1: X1 c= [:X1,X2,X3,X4,X5,X6,X7,X8:] or X1 c= [:X2,X3,X4,X5,X6,X7,X8,X1
:] or X1 c= [:X3,X4,X5,X6,X7,X8,X1,X2:] or X1 c= [:X4,X5,X6,X7,X8,X1,X2,X3:] or
X1 c= [:X5,X6,X7,X8,X1,X2,X3,X4:] or X1 c= [:X6,X7,X8,X1,X2,X3,X4,X5:] or X1 c=
[:X7,X8,X1,X2,X3,X4,X5,X6:] or X1 c= [:X8,X1,X2,X3,X4,X5,X6,X7:] and
A2: X1 <> {};
A3: [:X1,X2,X3,X4,X5,X6,X7,X8:]<>{} or [:X2,X3,X4,X5,X6,X7,X8,X1:]<>{} or [:
X3,X4,X5,X6,X7,X8,X1,X2:]<>{} or [:X4,X5,X6,X7,X8,X1,X2,X3:]<>{} or [:X5,X6,X7,
X8,X1,X2,X3,X4:]<>{} or [:X6,X7,X8,X1,X2,X3,X4,X5:]<>{} or [:X7,X8,X1,X2,X3,X4,
X5,X6:]<>{} or [:X8,X1,X2,X3,X4,X5,X6,X7:]<>{} by A1,A2,XBOOLE_1:3;
then
A4: X4<>{} & X5<>{} by Th132;
A5: X8<> {} by A3,Th132;
A6: X6<>{} & X7<>{} by A3,Th132;
A7: X2<>{} & X3<>{} by A3,Th132;
now
per cases by A1;
suppose
A8: X1 c= [:X1,X2,X3,X4,X5,X6,X7,X8:];
consider y such that
A9: y in X1 and
A10: for x1,x2,x3,x4,x5,x6,x7,x8 st x1 in X1 or x2 in X1 holds y <>
[x1,x2,x3,x4,x5,x6,x7,x8] by A2,Th125;
reconsider y as Element of [:X1,X2,X3,X4,X5,X6,X7,X8:] by A8,A9;
y = [y`1,y`2,y`3,y`4,y`5,y`6,y`7,y`8] by A2,A7,A4,A6,A5,Th138;
hence contradiction by A2,A10;
end;
suppose
X1 c= [:X2,X3,X4,X5,X6,X7,X8,X1:];
then X1 c= [:[:X2,X3:],X4,X5,X6,X7,X8,X1:] by Th12;
hence thesis by A2,Th98;
end;
suppose
X1 c= [:X3,X4,X5,X6,X7,X8,X1,X2:];
then X1 c= [:[:X3,X4:],X5,X6,X7,X8,X1,X2:] by Th12;
hence thesis by A2,Th98;
end;
suppose
X1 c= [:X4,X5,X6,X7,X8,X1,X2,X3:];
then X1 c= [:[:X4,X5:],X6,X7,X8,X1,X2,X3:] by Th12;
hence thesis by A2,Th98;
end;
suppose
X1 c= [:X5,X6,X7,X8,X1,X2,X3,X4:];
then X1 c= [:[:X5,X6:],X7,X8,X1,X2,X3,X4:] by Th12;
hence thesis by A2,Th98;
end;
suppose
X1 c= [:X6,X7,X8,X1,X2,X3,X4,X5:];
then X1 c= [:[:X6,X7:],X8,X1,X2,X3,X4,X5:] by Th12;
hence thesis by A2,Th98;
end;
suppose
X1 c= [:X7,X8,X1,X2,X3,X4,X5,X6:];
then X1 c= [:[:X7,X8:],X1,X2,X3,X4,X5,X6:] by Th12;
hence thesis by A2,Th98;
end;
suppose
A11: X1 c= [:X8,X1,X2,X3,X4,X5,X6,X7:];
consider y such that
A12: y in X1 and
A13: for x1,x2,x3,x4,x5,x6,x7,x8 st x1 in X1 or x2 in X1 holds y <>
[x1,x2,x3,x4,x5,x6,x7,x8] by A2,Th125;
reconsider y as Element of [:X8,X1,X2,X3,X4,X5,X6,X7:] by A11,A12;
y = [y`1,y`2,y`3,y`4,y`5,y`6,y`7,y`8] by A2,A7,A4,A6,A5,Th138;
hence thesis by A2,A13;
end;
end;
hence contradiction;
end;
theorem Th141:
[:X1,X2,X3,X4,X5,X6,X7,X8:] meets [:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8:]
implies X1 meets Y1 & X2 meets Y2 & X3 meets Y3 & X4 meets Y4 & X5 meets Y5 &
X6 meets Y6 & X7 meets Y7 & X8 meets Y8
proof
assume
A1: [:X1,X2,X3,X4,X5,X6,X7,X8:] meets [:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8:];
A2: [:[:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:],X7:] = [:X1,X2,X3,X4,X5,X6,X7:]
& [: [:[:[:[:[:Y1,Y2:],Y3:],Y4:],Y5:],Y6:],Y7:] = [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:]
by Th47;
[:[:[:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:],X7:],X8:] = [:X1,X2,X3,X4,X5,
X6,X7,X8:] & [:[:[:[:[:[:[:Y1,Y2:],Y3:],Y4:],Y5:],Y6:],Y7:],Y8:] = [:Y1,Y2,Y3,
Y4,Y5, Y6,Y7,Y8:] by Th126;
then
[:[:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:],X7:] meets [:[:[:[:[:[:Y1,Y2:],
Y3:],Y4:],Y5:],Y6:],Y7:] by A1,ZFMISC_1:104;
hence thesis by A2,A1,Th99,ZFMISC_1:104;
end;
theorem Th142:
[:{x1},{x2},{x3},{x4},{x5},{x6},{x7},{x8}:] = { [x1,x2,x3,x4,x5 ,x6,x7,x8] }
proof
thus [:{x1},{x2},{x3},{x4},{x5},{x6},{x7},{x8}:] = [:[:{x1},{x2}:],{x3},{x4}
,{x5},{x6},{x7},{x8}:] by Th12
.= [:{[x1,x2]}, {x3},{x4},{x5},{x6},{x7},{x8}:] by ZFMISC_1:29
.= { [[x1,x2], x3, x4, x5, x6, x7, x8]} by Th100
.= { [x1,x2,x3,x4,x5,x6,x7,x8] } by Th6;
end;
:: 8 - Tuples
reserve x for Element of [:X1,X2,X3,X4,X5,X6,X7,X8:];
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8<>{}
implies for x1,x2,x3,x4,x5,x6,x7,x8 st x = [x1,x2,x3,x4,x5,x6,x7,x8] holds x`1
= x1 & x`2 = x2 & x`3 = x3 & x`4 = x4 & x`5 = x5 & x`6 = x6 & x`7 = x7 & x`8 =
x8 by Def27,Def28,Def29,Def30,Def31,Def32,Def33,Def34;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8<>{}
& (for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8]
holds y1 = xx1) implies y1 =x`1
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <> {} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,
xx7,xx8] holds y1 = xx1;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7,x`8] by A1,Th138;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8<>{}
& (for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8]
holds y2 = xx2) implies y2 =x`2
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <> {} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,
xx7,xx8] holds y2 = xx2;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7,x`8] by A1,Th138;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8<>{}
& (for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8]
holds y3 = xx3) implies y3 =x`3
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <> {} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,
xx7,xx8] holds y3 = xx3;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7,x`8] by A1,Th138;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8<>{}
& (for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8]
holds y4 = xx4) implies y4 =x`4
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <> {} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,
xx7,xx8] holds y4 = xx4;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7,x`8] by A1,Th138;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8<>{}
& (for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8]
holds y5 = xx5) implies y5 =x`5
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <> {} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,
xx7,xx8] holds y5 = xx5;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7,x`8] by A1,Th138;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8<>{}
& (for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8]
holds y6 = xx6) implies y6 =x`6
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <> {} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,
xx7,xx8] holds y6 = xx6;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7,x`8] by A1,Th138;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8<>{}
& (for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8]
holds y7 = xx7) implies y7 =x`7
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <> {} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,
xx7,xx8] holds y7 = xx7;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7,x`8] by A1,Th138;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8<>{}
& (for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8]
holds y8 = xx8) implies y8 =x`8
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <> {} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 st x = [xx1,xx2,xx3,xx4,xx5,xx6,
xx7,xx8] holds y8 = xx8;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7,x`8] by A1,Th138;
hence thesis by A2;
end;
theorem Th152:
y in [: X1,X2,X3,X4,X5,X6,X7,X8 :] implies ex x1,x2,x3,x4,x5,x6
,x7,x8 st x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7
in X7 & x8 in X8 & y = [x1,x2,x3,x4,x5,x6,x7,x8]
proof
assume y in [: X1,X2,X3,X4,X5,X6,X7,X8 :];
then consider x1234567, x8 being set such that
A1: x1234567 in [:X1,X2,X3,X4,X5,X6,X7:] and
A2: x8 in X8 and
A3: y = [x1234567,x8] by ZFMISC_1:def 2;
consider x1, x2, x3, x4, x5, x6, x7 such that
A4: x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7
in X7 and
A5: x1234567 = [x1,x2,x3,x4,x5,x6,x7] by A1,Th109;
y = [x1,x2,x3,x4,x5,x6,x7,x8] by A3,A5;
hence thesis by A2,A4;
end;
theorem Th153:
[x1,x2,x3,x4,x5,x6,x7,x8] in [: X1,X2,X3,X4,X5,X6,X7,X8 :] iff
x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7 in X7 & x8
in X8
proof
A1: [x1,x2] in [:X1,X2:] iff x1 in X1 & x2 in X2 by ZFMISC_1:87;
[:X1,X2,X3,X4,X5,X6,X7,X8:] = [:[:X1,X2:],X3,X4,X5,X6,X7,X8:] & [x1,x2,
x3,x4,x5,x6,x7,x8] = [[x1,x2],x3,x4,x5,x6,x7,x8] by Th6,Th12;
hence thesis by A1,Th110;
end;
theorem
(for y holds y in Z iff ex x1,x2,x3,x4,x5,x6,x7,x8 st x1 in X1 & x2 in
X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7 in X7 & x8 in X8 & y = [x1,
x2,x3,x4,x5,x6,x7,x8]) implies Z = [: X1,X2,X3,X4,X5,X6,X7,X8 :]
proof
assume
A1: for y holds y in Z iff ex x1,x2,x3,x4,x5,x6,x7,x8 st x1 in X1 & x2
in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7 in X7 & x8 in X8 & y = [
x1,x2,x3,x4,x5,x6,x7,x8];
now
let y;
thus y in Z implies y in [:[:X1,X2,X3,X4,X5,X6,X7:],X8:]
proof
assume y in Z;
then consider x1,x2,x3,x4,x5,x6,x7,x8 such that
A2: x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6
& x7 in X7 and
A3: x8 in X8 & y = [x1,x2,x3,x4,x5,x6,x7,x8] by A1;
[x1,x2,x3,x4,x5,x6,x7] in [:X1,X2,X3,X4,X5,X6,X7:] by A2,Th110;
hence thesis by A3,ZFMISC_1:def 2;
end;
assume y in [:[:X1,X2,X3,X4,X5,X6,X7:],X8:];
then consider x1234567,x8 being set such that
A4: x1234567 in [:X1,X2,X3,X4,X5,X6,X7:] and
A5: x8 in X8 and
A6: y = [x1234567,x8] by ZFMISC_1:def 2;
consider x1,x2,x3,x4,x5,x6,x7 such that
A7: x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 &
x7 in X7 and
A8: x1234567 = [x1,x2,x3,x4,x5,x6,x7] by A4,Th109;
y = [x1,x2,x3,x4,x5,x6,x7,x8] by A6,A8;
hence y in Z by A1,A5,A7;
end;
hence thesis by TARSKI:1;
end;
theorem Th155:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} &
X8<>{} & Y1<>{} & Y2<>{} & Y3<>{} & Y4<>{} & Y5<>{} & Y6<>{} & Y7<>{} & Y8<>{}
implies for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:], y being Element of
[:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8:] holds x = y implies x`1 = y`1 & x`2 = y`2 & x`3 = y
`3 & x`4 = y`4 & x`5 = y`5 & x`6 = y`6 & x`7 = y`7 & x`8 = y`8
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <> {} and
A2: Y1<>{} & Y2<>{} & Y3<>{} & Y4<>{} & Y5<>{} & Y6<>{} & Y7<>{} & Y8 <> {};
let x be Element of [:X1,X2,X3,X4,X5,X6,X7,X8:];
let y be Element of [:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8:];
assume
A3: x = y;
thus x`1 = (x qua set)`1`1`1`1`1`1`1 by A1,Th139
.= y`1 by A2,A3,Th139;
thus x`2 = (x qua set)`1`1`1`1`1`1`2 by A1,Th139
.= y`2 by A2,A3,Th139;
thus x`3 = (x qua set)`1`1`1`1`1`2 by A1,Th139
.= y`3 by A2,A3,Th139;
thus x`4 = (x qua set)`1`1`1`1`2 by A1,Th139
.= y`4 by A2,A3,Th139;
thus x`5 = (x qua set)`1`1`1`2 by A1,Th139
.= y`5 by A2,A3,Th139;
thus x`6 = (x qua set)`1`1`2 by A1,Th139
.= y`6 by A2,A3,Th139;
thus x`7 = (x qua set)`1`2 by A1,Th139
.= y`7 by A2,A3,Th139;
thus x`8 = (x qua set)`2 by A1,Th139
.= y`8 by A2,A3,Th139;
end;
theorem
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8:] st x in [:A1,A2,A3,
A4,A5,A6,A7,A8:] holds x`1 in A1 & x`2 in A2 & x`3 in A3 & x`4 in A4 & x`5 in
A5 & x`6 in A6 & x`7 in A7 & x`8 in A8
proof
let x be Element of [:X1,X2,X3,X4,X5,X6,X7,X8:];
assume
A1: x in [:A1,A2,A3,A4,A5,A6,A7,A8:];
then reconsider y = x as Element of [:A1,A2,A3,A4,A5,A6,A7,A8:];
A2<>{} by A1,Th132;
then
A2: y`2 in A2;
A6<>{} by A1,Th132;
then
A3: y`6 in A6;
A5<>{} by A1,Th132;
then
A4: y`5 in A5;
A4<>{} by A1,Th132;
then
A5: y`4 in A4;
A3<>{} by A1,Th132;
then
A6: y`3 in A3;
A8<> {} by A1,Th132;
then
A7: y`8 in A8;
A7<>{} by A1,Th132;
then
A8: y`7 in A7;
A1<>{} by A1,Th132;
then y`1 in A1;
hence thesis by A2,A6,A5,A4,A3,A8,A7,Th155;
end;
theorem Th157:
X1 c= Y1 & X2 c= Y2 & X3 c= Y3 & X4 c= Y4 & X5 c= Y5 & X6 c= Y6
& X7 c= Y7 & X8 c= Y8 implies [:X1,X2,X3,X4,X5,X6,X7,X8:] c= [:Y1,Y2,Y3,Y4,Y5,
Y6,Y7,Y8:]
proof
assume
X1 c= Y1 & X2 c= Y2 & X3 c= Y3 & X4 c= Y4 & X5 c= Y5 & X6 c= Y6 & X7 c= Y7;
then
A1: [:X1,X2,X3,X4,X5,X6,X7:] c= [:Y1,Y2,Y3,Y4,Y5,Y6,Y7:] by Th114;
assume X8 c= Y8;
hence thesis by A1,ZFMISC_1:96;
end;
theorem
[:A1,A2,A3,A4,A5,A6,A7,A8:] is Subset of [:X1,X2,X3,X4,X5,X6,X7,X8:]
by Th157;
begin :: Original MCART_6
theorem
X <> {} implies ex Y st Y in X & for Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB,
YC,YD,YE st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 &
Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC & YC in YD & YD in YE & YE
in Y holds Y1 misses X
proof
defpred P1[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB,YC,YD st Y1 in Y2
& Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 &
Y9 in YA & YA in YB & YB in YC & YC in YD & YD in $1 & Y1 meets X;
consider Z1 such that
A1: for Y holds Y in Z1 iff Y in union X & P1[Y] from XBOOLE_0:sch 1;
defpred P14[set] means $1 meets X;
defpred P13[set] means ex Y1 st Y1 in $1 & Y1 meets X;
defpred P12[set] means ex Y1,Y2 st Y1 in Y2 & Y2 in $1 & Y1 meets X;
defpred P11[set] means ex Y1,Y2,Y3 st Y1 in Y2 & Y2 in Y3 & Y3 in $1 & Y1
meets X;
defpred P10[set] means ex Y1,Y2,Y3,Y4 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4
in $1 & Y1 meets X;
defpred P9[set] means ex Y1,Y2,Y3,Y4,Y5 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 &
Y4 in Y5 & Y5 in $1 & Y1 meets X;
defpred P8[set] means ex Y1,Y2,Y3,Y4,Y5,Y6 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4
& Y4 in Y5 & Y5 in Y6 & Y6 in $1 & Y1 meets X;
defpred P7[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7 st Y1 in Y2 & Y2 in Y3 & Y3 in
Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in $1 & Y1 meets X;
defpred P6[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8 st Y1 in Y2 & Y2 in Y3 & Y3
in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in $1 & Y1 meets X;
defpred P5[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9 st Y1 in Y2 & Y2 in Y3 &
Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in $1 & Y1
meets X;
defpred P4[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA st Y1 in Y2 & Y2 in
Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA
& YA in $1 & Y1 meets X;
defpred P3[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB st Y1 in Y2 & Y2
in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in
YA & YA in YB & YB in $1 & Y1 meets X;
defpred P2[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB,YC st Y1 in Y2 &
Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9
in YA & YA in YB & YB in YC & YC in $1 & Y1 meets X;
consider Z2 such that
A2: for Y holds Y in Z2 iff Y in union union X & P2[Y] from XBOOLE_0:sch
1;
consider Z7 such that
A3: for Y holds Y in Z7 iff Y in union union union union union union
union X & P7[Y] from XBOOLE_0:sch 1;
consider Z6 such that
A4: for Y holds Y in Z6 iff Y in union union union union union union X &
P6[Y] from XBOOLE_0:sch 1;
consider ZE such that
A5: for Y holds Y in ZE iff Y in union union union union union union
union union union union (union union union union X) & P14[Y] from XBOOLE_0:sch
1;
consider Z3 such that
A6: for Y holds Y in Z3 iff Y in union union union X & P3[Y] from
XBOOLE_0:sch 1;
consider Z5 such that
A7: for Y holds Y in Z5 iff Y in union union union union union X & P5[Y]
from XBOOLE_0:sch 1;
consider Z4 such that
A8: for Y holds Y in Z4 iff Y in union union union union X & P4[Y] from
XBOOLE_0:sch 1;
consider ZD such that
A9: for Y holds Y in ZD iff Y in union union union union union union
union union union union (union union union X) & P13[Y] from XBOOLE_0:sch 1;
consider ZC such that
A10: for Y holds Y in ZC iff Y in union union union union union union
union union union union (union union X) & P12[Y] from XBOOLE_0:sch 1;
consider ZB such that
A11: for Y holds Y in ZB iff Y in union union union union union union
union union union union (union X) & P11[Y] from XBOOLE_0:sch 1;
consider ZA such that
A12: for Y holds Y in ZA iff Y in union union union union union union
union union union union X & P10[Y] from XBOOLE_0:sch 1;
consider Z9 such that
A13: for Y holds Y in Z9 iff Y in union union union union union union
union union union X & P9[Y] from XBOOLE_0:sch 1;
consider Z8 such that
A14: for Y holds Y in Z8 iff Y in union union union union union union
union union X & P8[Y] from XBOOLE_0:sch 1;
set V = (X \/ Z1 \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB \/ ZC \/ ZD \/ ZE;
A15: V = (X \/ (Z1 \/ Z2) \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB \/ ZC \/ ZD \/ ZE by XBOOLE_1:4
.= (X \/ (Z1 \/ Z2 \/ Z3) \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB \/ ZC \/ ZD \/ ZE by XBOOLE_1:4
.= (X \/ (Z1 \/ Z2 \/ Z3 \/ Z4)) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB \/ ZC \/ ZD \/ ZE by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5) \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB
\/ ZC \/ ZD \/ ZE by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6) \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB
\/ ZC \/ ZD \/ ZE by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7) \/ Z8 \/ Z9 \/ ZA \/ ZB
\/ ZC \/ ZD \/ ZE by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8) \/ Z9 \/ ZA \/ ZB
\/ ZC \/ ZD \/ ZE by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9) \/ ZA \/ ZB
\/ ZC \/ ZD \/ ZE by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA) \/ ZB
\/ ZC \/ ZD \/ ZE by XBOOLE_1:4
.= X \/ ((Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB) \/ ZC \/ ZD \/ ZE by XBOOLE_1:4
.= X \/ ((Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB \/ ZC) \/ ZD \/ ZE by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB
\/ ZC \/ ZD) \/ ZE by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB
\/ ZC \/ ZD \/ ZE) by XBOOLE_1:4;
assume X <> {};
then consider Y such that
A16: Y in V and
A17: Y misses V by MCART_1:1;
Y in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB \/ ZC \/ ZD or Y in ZE by A16,XBOOLE_0:def 3;
then Y in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB \/ ZC or Y in ZD or Y in ZE by XBOOLE_0:def 3;
then Y in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB or Y in ZC or Y in ZD or Y in ZE by XBOOLE_0:def 3;
then
Y in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA or Y
in ZB or Y in ZC or Y in ZD or Y in ZE by XBOOLE_0:def 3;
then
Y in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 or Y in ZA
or Y in ZB or Y in ZC or Y in ZD or Y in ZE by XBOOLE_0:def 3;
then Y in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 or Y in Z9 or Y
in ZA or Y in ZB or Y in ZC or Y in ZD or Y in ZE by XBOOLE_0:def 3;
then Y in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 or Y in Z8 or Y in Z9
or Y in ZA or Y in ZB or Y in ZC or Y in ZD or Y in ZE by XBOOLE_0:def 3;
then Y in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 or Y in Z7 or Y in Z8 or Y
in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD or Y in ZE by
XBOOLE_0:def 3;
then
Y in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 or Y in Z6 or Y in Z7 or Y in Z8
or Y in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD or Y in ZE by
XBOOLE_0:def 3;
then Y in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 or Y in Z5 or Y in Z6 or Y in Z7 or Y
in Z8 or Y in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD or Y in ZE by
XBOOLE_0:def 3;
then
Y in X \/ Z1 \/ Z2 \/ Z3 or Y in Z4 or Y in Z5 or Y in Z6 or Y in Z7 or
Y in Z8 or Y in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD or Y in ZE by
XBOOLE_0:def 3;
then Y in X \/ Z1 \/ Z2 or Y in Z3 or Y in Z4 or Y in Z5 or Y in Z6 or Y in
Z7 or Y in Z8 or Y in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD or Y in ZE
by XBOOLE_0:def 3;
then
A18: Y in X \/ Z1 or Y in Z2 or Y in Z3 or Y in Z4 or Y in Z5 or Y in Z6 or
Y in Z7 or Y in Z8 or Y in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD or Y
in ZE by XBOOLE_0:def 3;
assume
A19: not thesis;
per cases by A18,XBOOLE_0:def 3;
suppose
A20: Y in X;
set Z15 = X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5;
consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB,YC,YD,YE such that
A21: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 &
Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC & YC in YD & YD in YE
and
A22: YE in Y and
A23: not Y1 misses X by A19,A20;
YE in union X by A20,A22,TARSKI:def 4;
then YE in Z1 by A1,A21,A23;
then YE in X \/ Z1 by XBOOLE_0:def 3;
then YE in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
then YE in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 3;
then YE in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then YE in Z15 by XBOOLE_0:def 3;
then Y meets Z15 by A22,XBOOLE_0:3;
then Y meets Z15 \/ Z6 by XBOOLE_1:70;
then Y meets Z15 \/ Z6 \/ Z7 by XBOOLE_1:70;
then Y meets Z15 \/ Z6 \/ Z7 \/ Z8 by XBOOLE_1:70;
then Y meets Z15 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by XBOOLE_1:70;
then Y meets Z15 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA by XBOOLE_1:70;
then Y meets Z15 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB by XBOOLE_1:70;
then Y meets Z15 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB \/ ZC by XBOOLE_1:70;
then Y meets Z15 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB \/ ZC \/ ZD by
XBOOLE_1:70;
hence contradiction by A17,XBOOLE_1:70;
end;
suppose
A24: Y in Z1;
set Z15 = X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5;
consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB,YC,YD such that
A25: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 &
Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC & YC in YD and
A26: YD in Y and
A27: Y1 meets X by A1,A24;
Y in union X by A1,A24;
then YD in union union X by A26,TARSKI:def 4;
then YD in Z2 by A2,A25,A27;
then YD in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
then Y meets X \/ Z1 \/ Z2 by A26,XBOOLE_0:3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_1:70;
then Y meets Z15 by XBOOLE_1:70;
then Y meets Z15 \/ Z6 by XBOOLE_1:70;
then Y meets Z15 \/ Z6 \/ Z7 by XBOOLE_1:70;
then Y meets Z15 \/ Z6 \/ Z7 \/ Z8 by XBOOLE_1:70;
then Y meets Z15 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by XBOOLE_1:70;
then Y meets Z15 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA by XBOOLE_1:70;
then Y meets Z15 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB by XBOOLE_1:70;
then Y meets Z15 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB \/ ZC by XBOOLE_1:70;
then Y meets Z15 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB \/ ZC \/ ZD by
XBOOLE_1:70;
hence contradiction by A17,XBOOLE_1:70;
end;
suppose
A28: Y in Z2;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB,YC such that
A29: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7
& Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC and
A30: YC in Y and
A31: Y1 meets X by A2;
Y in union union X by A2,A28;
then YC in union union union X by A30,TARSKI:def 4;
then YC in Z3 by A6,A29,A31;
then YC in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 3;
then YC in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then YC in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then YC in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then YC in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then YC in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then YC in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then YC in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then YC in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then YC in (X \/ Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_0:def 3;
then YC in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD by XBOOLE_0:def 3;
then YC in V by XBOOLE_0:def 3;
hence contradiction by A17,A30,XBOOLE_0:3;
end;
suppose
A32: Y in Z3;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB such that
A33: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7
& Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB and
A34: YB in Y and
A35: Y1 meets X by A6;
Y in union union union X by A6,A32;
then YB in union union union union X by A34,TARSKI:def 4;
then YB in Z4 by A8,A33,A35;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB by XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB \/ ZC by XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB \/ ZC \/ ZD by XBOOLE_0:def 3;
then YB in V by XBOOLE_0:def 3;
hence contradiction by A17,A34,XBOOLE_0:3;
end;
suppose
A36: Y in Z4;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA such that
A37: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7
& Y7 in Y8 & Y8 in Y9 & Y9 in YA and
A38: YA in Y and
A39: Y1 meets X by A8;
Y in union union union union X by A8,A36;
then YA in union union union union union X by A38,TARSKI:def 4;
then YA in Z5 by A7,A37,A39;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then YA in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then YA in (X \/ Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_0:def 3;
then YA in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD by XBOOLE_0:def 3;
then YA in V by XBOOLE_0:def 3;
hence contradiction by A17,A38,XBOOLE_0:3;
end;
suppose
A40: Y in Z5;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9 such that
A41: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7
& Y7 in Y8 & Y8 in Y9 and
A42: Y9 in Y and
A43: Y1 meets X by A7;
Y in union union union union union X by A7,A40;
then Y9 in union union union union union union X by A42,TARSKI:def 4;
then Y9 in Z6 by A4,A41,A43;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y9 in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then Y9 in (X \/ Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_0:def 3;
then Y9 in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD by XBOOLE_0:def 3;
then Y9 in V by XBOOLE_0:def 3;
hence contradiction by A17,A42,XBOOLE_0:3;
end;
suppose
A44: Y in Z6;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8 such that
A45: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7
& Y7 in Y8 and
A46: Y8 in Y and
A47: Y1 meets X by A4;
Y in union union union union union union X by A4,A44;
then Y8 in union union union union union union union X by A46,TARSKI:def 4;
then Y8 in Z7 by A3,A45,A47;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y8 in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB \/ ZC by XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB \/ ZC \/ ZD by XBOOLE_0:def 3;
then Y8 in V by XBOOLE_0:def 3;
hence contradiction by A17,A46,XBOOLE_0:3;
end;
suppose
A48: Y in Z7;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7 such that
A49: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 and
A50: Y7 in Y and
A51: Y1 meets X by A3;
Y in union union union union union union union X by A3,A48;
then
Y7 in union union union union union union union union X by A50,TARSKI:def 4;
then Y7 in Z8 by A14,A49,A51;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y7 in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then Y7 in (X \/ Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_0:def 3;
then Y7 in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD by XBOOLE_0:def 3;
then Y7 in V by XBOOLE_0:def 3;
hence contradiction by A17,A50,XBOOLE_0:3;
end;
suppose
A52: Y in Z8;
then consider Y1,Y2,Y3,Y4,Y5,Y6 such that
A53: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 and
A54: Y6 in Y and
A55: Y1 meets X by A14;
Y in union union union union union union union union X by A14,A52;
then Y6 in union union union union union union union union union X by A54,
TARSKI:def 4;
then Y6 in Z9 by A13,A53,A55;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y6 in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then Y6 in ((X \/ Z1) \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_0:def 3;
then Y6 in ((X \/ Z1) \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD by XBOOLE_0:def 3;
then Y6 in V by XBOOLE_0:def 3;
hence contradiction by A17,A54,XBOOLE_0:3;
end;
suppose
A56: Y in Z9;
then consider Y1,Y2,Y3,Y4,Y5 such that
A57: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 and
A58: Y5 in Y and
A59: Y1 meets X by A13;
Y in union union union union union union union union union X by A13,A56;
then Y5 in union union union union union union union union union union X
by A58,TARSKI:def 4;
then Y5 in ZA by A12,A57,A59;
then Y5 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y5 in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then Y5 in (X \/ Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_0:def 3;
then Y5 in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD by XBOOLE_0:def 3;
then Y5 in V by XBOOLE_0:def 3;
hence contradiction by A17,A58,XBOOLE_0:3;
end;
suppose
A60: Y in ZA;
then consider Y1,Y2,Y3,Y4 such that
A61: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 and
A62: Y4 in Y and
A63: Y1 meets X by A12;
Y in union union union union union union union union union union X
by A12,A60;
then Y4 in union union union union union union union union union union
union X by A62,TARSKI:def 4;
then Y4 in ZB by A11,A61,A63;
then Y4 in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then Y4 in (X \/ Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_0:def 3;
then Y4 in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD by XBOOLE_0:def 3;
then Y4 in V by XBOOLE_0:def 3;
hence contradiction by A17,A62,XBOOLE_0:3;
end;
suppose
A64: Y in ZB;
then consider Y1,Y2,Y3 such that
A65: Y1 in Y2 & Y2 in Y3 and
A66: Y3 in Y and
A67: Y1 meets X by A11;
Y in union union union union union union union union union union (
union X) by A11,A64;
then Y3 in union union union union union union union union union union
union union X by A66,TARSKI:def 4;
then Y3 in ZC by A10,A65,A67;
then Y3 in ((X \/ Z1) \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_0:def 3;
then Y3 in ((X \/ Z1) \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD by XBOOLE_0:def 3;
then Y3 in V by XBOOLE_0:def 3;
hence contradiction by A17,A66,XBOOLE_0:3;
end;
suppose
A68: Y in ZC;
then consider Y1,Y2 such that
A69: Y1 in Y2 and
A70: Y2 in Y and
A71: Y1 meets X by A10;
Y in union union union union union union union union union union
union union X by A10,A68;
then Y2 in union union union union union union union union union union
union union union X by A70,TARSKI:def 4;
then Y2 in ZD by A9,A69,A71;
then Y2 in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD by XBOOLE_0:def 3;
then Y2 in V by XBOOLE_0:def 3;
hence contradiction by A17,A70,XBOOLE_0:3;
end;
suppose
A72: Y in ZD;
then consider Y1 such that
A73: Y1 in Y and
A74: Y1 meets X by A9;
Y in union union union union union union union union union union
union union union X by A9,A72;
then Y1 in union union union union union union union union union union
union union union union X by A73,TARSKI:def 4;
then Y1 in ZE by A5,A74;
then Y1 in V by XBOOLE_0:def 3;
hence contradiction by A17,A73,XBOOLE_0:3;
end;
suppose
Y in ZE;
then Y meets X by A5;
hence contradiction by A15,A17,XBOOLE_1:70;
end;
end;
theorem
X <> {} implies ex Y st Y in X & for Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB,
YC,YD,YE,YF st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7
& Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC & YC in YD & YD in YE &
YE in YF & YF in Y holds Y1 misses X
proof
defpred P1[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB,YC,YD,YE st Y1 in
Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9
& Y9 in YA & YA in YB & YB in YC & YC in YD & YD in YE & YE in $1 & Y1 meets X;
consider Z1 such that
A1: for Y holds Y in Z1 iff Y in union X & P1[Y] from XBOOLE_0:sch 1;
defpred P15[set] means $1 meets X;
defpred P14[set] means ex Y1 st Y1 in $1 & Y1 meets X;
defpred P13[set] means ex Y1,Y2 st Y1 in Y2 & Y2 in $1 & Y1 meets X;
defpred P12[set] means ex Y1,Y2,Y3 st Y1 in Y2 & Y2 in Y3 & Y3 in $1 & Y1
meets X;
defpred P11[set] means ex Y1,Y2,Y3,Y4 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4
in $1 & Y1 meets X;
defpred P10[set] means ex Y1,Y2,Y3,Y4,Y5 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4 &
Y4 in Y5 & Y5 in $1 & Y1 meets X;
defpred P9[set] means ex Y1,Y2,Y3,Y4,Y5,Y6 st Y1 in Y2 & Y2 in Y3 & Y3 in Y4
& Y4 in Y5 & Y5 in Y6 & Y6 in $1 & Y1 meets X;
defpred P8[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7 st Y1 in Y2 & Y2 in Y3 & Y3 in
Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in $1 & Y1 meets X;
defpred P7[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8 st Y1 in Y2 & Y2 in Y3 & Y3
in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in $1 & Y1 meets X;
defpred P6[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9 st Y1 in Y2 & Y2 in Y3 &
Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in $1 & Y1
meets X;
defpred P5[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA st Y1 in Y2 & Y2 in
Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in YA
& YA in $1 & Y1 meets X;
defpred P4[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB st Y1 in Y2 & Y2
in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9 in
YA & YA in YB & YB in $1 & Y1 meets X;
defpred P3[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB,YC st Y1 in Y2 &
Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 & Y9
in YA & YA in YB & YB in YC & YC in $1 & Y1 meets X;
defpred P2[set] means ex Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB,YC,YD st Y1 in Y2
& Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 & Y7 in Y8 & Y8 in Y9 &
Y9 in YA & YA in YB & YB in YC & YC in YD & YD in $1 & Y1 meets X;
consider Z2 such that
A2: for Y holds Y in Z2 iff Y in union union X & P2[Y] from XBOOLE_0:sch
1;
consider Z9 such that
A3: for Y holds Y in Z9 iff Y in union union union union union union
union union union X & P9[Y] from XBOOLE_0:sch 1;
consider Z8 such that
A4: for Y holds Y in Z8 iff Y in union union union union union union
union union X & P8[Y] from XBOOLE_0:sch 1;
consider ZF such that
A5: for Y holds Y in ZF iff Y in union union union union union union
union union union union (union union union union union X) & P15[Y] from
XBOOLE_0:sch 1;
consider Z3 such that
A6: for Y holds Y in Z3 iff Y in union union union X & P3[Y] from
XBOOLE_0:sch 1;
consider Z7 such that
A7: for Y holds Y in Z7 iff Y in union union union union union union
union X & P7[Y] from XBOOLE_0:sch 1;
consider Z6 such that
A8: for Y holds Y in Z6 iff Y in union union union union union union X &
P6[Y] from XBOOLE_0:sch 1;
consider ZE such that
A9: for Y holds Y in ZE iff Y in union union union union union union
union union union union union union union union X & P14[Y] from XBOOLE_0:sch 1;
consider Z5 such that
A10: for Y holds Y in Z5 iff Y in union union union union union X & P5[Y]
from XBOOLE_0:sch 1;
consider Z4 such that
A11: for Y holds Y in Z4 iff Y in union union union union X & P4[Y] from
XBOOLE_0:sch 1;
consider ZD such that
A12: for Y holds Y in ZD iff Y in union union union union union union
union union union union union union union X & P13[Y] from XBOOLE_0:sch 1;
consider ZC such that
A13: for Y holds Y in ZC iff Y in union union union union union union
union union union union union union X & P12[Y] from XBOOLE_0:sch 1;
consider ZB such that
A14: for Y holds Y in ZB iff Y in union union union union union union
union union union union union X & P11[Y] from XBOOLE_0:sch 1;
consider ZA such that
A15: for Y holds Y in ZA iff Y in union union union union union union
union union union union X & P10[Y] from XBOOLE_0:sch 1;
set V = X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB
\/ ZC \/ ZD \/ ZE \/ ZF;
A16: V = (X \/ (Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5) \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB \/ ZC \/ ZD \/ ZE \/ ZF by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB
\/ ZC \/ ZD \/ ZE \/ ZF by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB
\/ ZC \/ ZD \/ ZE \/ ZF by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5) \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB
\/ ZC \/ ZD \/ ZE \/ ZF by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6) \/ Z7 \/ Z8 \/ Z9 \/ ZA \/ ZB
\/ ZC \/ ZD \/ ZE \/ ZF by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7) \/ Z8 \/ Z9 \/ ZA \/ ZB
\/ ZC \/ ZD \/ ZE \/ ZF by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8) \/ Z9 \/ ZA \/ ZB
\/ ZC \/ ZD \/ ZE \/ ZF by XBOOLE_1:4
.= X \/ (Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9) \/ ZA \/ ZB
\/ ZC \/ ZD \/ ZE \/ ZF by XBOOLE_1:4
.= X \/ ((Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA) \/
ZB \/ ZC \/ ZD \/ ZE \/ ZF by XBOOLE_1:4
.= X \/ ((Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB) \/ ZC \/ ZD \/ ZE \/ ZF by XBOOLE_1:4
.= X \/ ((Z1 \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB \/ ZC) \/ ZD \/ ZE \/ ZF by XBOOLE_1:4
.= X \/ ((Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5) \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB \/ ZC \/ ZD) \/ ZE \/ ZF by XBOOLE_1:4
.= X \/ ((Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6) \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB \/ ZC \/ ZD \/ ZE) \/ ZF by XBOOLE_1:4
.= X \/ ((Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6) \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB \/ ZC \/ ZD \/ ZE \/ ZF) by XBOOLE_1:4;
assume X <> {};
then consider Y such that
A17: Y in V and
A18: Y misses V by MCART_1:1;
Y in (X \/ Z1 \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB \/ ZC \/ ZD \/ ZE or Y in ZF by A17,XBOOLE_0:def 3;
then
Y in (X \/ Z1 \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB \/ ZC \/ ZD or Y in ZE or Y in ZF by XBOOLE_0:def 3;
then
Y in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB \/ ZC or Y in ZD or Y in ZE or Y in ZF by XBOOLE_0:def 3;
then
Y in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA \/
ZB or Y in ZC or Y in ZD or Y in ZE or Y in ZF by XBOOLE_0:def 3;
then
Y in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA or
Y in ZB or Y in ZC or Y in ZD or Y in ZE or Y in ZF by XBOOLE_0:def 3;
then Y in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 or Y in
ZA or Y in ZB or Y in ZC or Y in ZD or Y in ZE or Y in ZF by XBOOLE_0:def 3;
then Y in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 or Y in Z9 or
Y in ZA or Y in ZB or Y in ZC or Y in ZD or Y in ZE or Y in ZF by
XBOOLE_0:def 3;
then Y in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 or Y in Z8 or Y in
Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD or Y in ZE or Y in ZF by
XBOOLE_0:def 3;
then
Y in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 or Y in Z7 or Y in Z8 or Y
in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD or Y in ZE or Y in ZF by
XBOOLE_0:def 3;
then
Y in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 or Y in Z6 or Y in Z7 or Y in Z8
or Y in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD or Y in ZE or Y in ZF
by XBOOLE_0:def 3;
then Y in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 or Y in Z5 or Y in Z6 or Y in Z7 or Y
in Z8 or Y in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD or Y in ZE or Y in
ZF by XBOOLE_0:def 3;
then Y in (X \/ Z1 \/ Z2 \/ Z3) or Y in Z4 or Y in Z5 or Y in Z6 or Y in Z7
or Y in Z8 or Y in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD or Y in ZE or
Y in ZF by XBOOLE_0:def 3;
then Y in (X \/ Z1) \/ Z2 or Y in Z3 or Y in Z4 or Y in Z5 or Y in Z6 or Y
in Z7 or Y in Z8 or Y in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD or Y in
ZE or Y in ZF by XBOOLE_0:def 3;
then
A19: Y in X \/ Z1 or Y in Z2 or Y in Z3 or Y in Z4 or Y in Z5 or Y in Z6 or
Y in Z7 or Y in Z8 or Y in Z9 or Y in ZA or Y in ZB or Y in ZC or Y in ZD or Y
in ZE or Y in ZF by XBOOLE_0:def 3;
assume
A20: not thesis;
per cases by A19,XBOOLE_0:def 3;
suppose
A21: Y in X;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB,YC,YD,YE,YF such that
A22: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 &
Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC & YC in YD & YD in YE & YE
in YF and
A23: YF in Y and
A24: not Y1 misses X by A20;
YF in union X by A21,A23,TARSKI:def 4;
then YF in Z1 by A1,A22,A24;
then YF in X \/ Z1 by XBOOLE_0:def 3;
then YF in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
then YF in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 3;
then YF in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then YF in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by A23,XBOOLE_0:3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_1:70;
then
Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_1:70;
then Y meets (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_1:70;
then Y meets (X \/ Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_1:70;
then Y meets (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD by XBOOLE_1:70;
then Y meets (X \/ Z1 \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD \/ ZE by XBOOLE_1:70;
hence contradiction by A18,XBOOLE_1:70;
end;
suppose
A25: Y in Z1;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB,YC,YD,YE such that
A26: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 &
Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC & YC in YD & YD in YE
and
A27: YE in Y and
A28: Y1 meets X by A1;
Y in union X by A1,A25;
then YE in union union X by A27,TARSKI:def 4;
then YE in Z2 by A2,A26,A28;
then YE in X \/ Z1 \/ Z2 by XBOOLE_0:def 3;
then Y meets X \/ Z1 \/ Z2 by A27,XBOOLE_0:3;
then Y meets X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_1:70;
then Y meets X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA by XBOOLE_1:70;
then Y meets (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_1:70;
then Y meets (X \/ Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_1:70;
then Y meets (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD by XBOOLE_1:70;
then Y meets (X \/ Z1 \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD \/ ZE by XBOOLE_1:70;
hence contradiction by A18,XBOOLE_1:70;
end;
suppose
A29: Y in Z2;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB,YC,YD such that
A30: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7
& Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC & YC in YD and
A31: YD in Y and
A32: Y1 meets X by A2;
Y in union union X by A2,A29;
then YD in union union union X by A31,TARSKI:def 4;
then YD in Z3 by A6,A30,A32;
then YD in X \/ Z1 \/ Z2 \/ Z3 by XBOOLE_0:def 3;
then YD in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then YD in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then YD in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then YD in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then YD in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then YD in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then YD in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then YD in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then YD in (X \/ Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_0:def 3;
then YD in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD by XBOOLE_0:def 3;
then YD in (X \/ Z1 \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD \/ ZE by XBOOLE_0:def 3;
then YD in V by XBOOLE_0:def 3;
hence contradiction by A18,A31,XBOOLE_0:3;
end;
suppose
A33: Y in Z3;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB,YC such that
A34: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7
& Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB & YB in YC and
A35: YC in Y and
A36: Y1 meets X by A6;
Y in union union union X by A6,A33;
then YC in union union union union X by A35,TARSKI:def 4;
then YC in Z4 by A11,A34,A36;
then YC in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 by XBOOLE_0:def 3;
then YC in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then YC in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then YC in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then YC in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then YC in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then YC in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then YC in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB by XBOOLE_0:def 3;
then YC in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB \/ ZC by XBOOLE_0:def 3;
then YC in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB \/ ZC \/ ZD by XBOOLE_0:def 3;
then YC in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB \/ ZC \/ ZD \/ ZE by XBOOLE_0:def 3;
then YC in V by XBOOLE_0:def 3;
hence contradiction by A18,A35,XBOOLE_0:3;
end;
suppose
A37: Y in Z4;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA,YB such that
A38: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7
& Y7 in Y8 & Y8 in Y9 & Y9 in YA & YA in YB and
A39: YB in Y and
A40: Y1 meets X by A11;
Y in union union union union X by A11,A37;
then YB in union union union union union X by A39,TARSKI:def 4;
then YB in Z5 by A10,A38,A40;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 by XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then YB in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB \/ ZC by XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB \/ ZC \/ ZD by XBOOLE_0:def 3;
then YB in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB \/ ZC \/ ZD \/ ZE by XBOOLE_0:def 3;
then YB in V by XBOOLE_0:def 3;
hence contradiction by A18,A39,XBOOLE_0:3;
end;
suppose
A41: Y in Z5;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9,YA such that
A42: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7
& Y7 in Y8 & Y8 in Y9 & Y9 in YA and
A43: YA in Y and
A44: Y1 meets X by A10;
Y in union union union union union X by A10,A41;
then YA in union union union union union union X by A43,TARSKI:def 4;
then YA in Z6 by A8,A42,A44;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 by XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB by XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB \/ ZC by XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB \/ ZC \/ ZD by XBOOLE_0:def 3;
then YA in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB \/ ZC \/ ZD \/ ZE by XBOOLE_0:def 3;
then YA in V by XBOOLE_0:def 3;
hence contradiction by A18,A43,XBOOLE_0:3;
end;
suppose
A45: Y in Z6;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9 such that
A46: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7
& Y7 in Y8 & Y8 in Y9 and
A47: Y9 in Y and
A48: Y1 meets X by A8;
Y in union union union union union union X by A8,A45;
then Y9 in union union union union union union union X by A47,TARSKI:def 4;
then Y9 in Z7 by A7,A46,A48;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 by XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB by XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB \/ ZC by XBOOLE_0:def 3;
then Y9 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB \/ ZC \/ ZD by XBOOLE_0:def 3;
then Y9 in (X \/ Z1 \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD \/ ZE by XBOOLE_0:def 3;
then Y9 in V by XBOOLE_0:def 3;
hence contradiction by A18,A47,XBOOLE_0:3;
end;
suppose
A49: Y in Z7;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8 such that
A50: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7
& Y7 in Y8 and
A51: Y8 in Y and
A52: Y1 meets X by A7;
Y in union union union union union union union X by A7,A49;
then
Y8 in union union union union union union union union X by A51,TARSKI:def 4;
then Y8 in Z8 by A4,A50,A52;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 by
XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB by XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB \/ ZC by XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB \/ ZC \/ ZD by XBOOLE_0:def 3;
then Y8 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
\/ ZB \/ ZC \/ ZD \/ ZE by XBOOLE_0:def 3;
then Y8 in V by XBOOLE_0:def 3;
hence contradiction by A18,A51,XBOOLE_0:3;
end;
suppose
A53: Y in Z8;
then consider Y1,Y2,Y3,Y4,Y5,Y6,Y7 such that
A54: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 & Y6 in Y7 and
A55: Y7 in Y and
A56: Y1 meets X by A4;
Y in union union union union union union union union X by A4,A53;
then Y7 in union union union union union union union union union X by A55,
TARSKI:def 4;
then Y7 in Z9 by A3,A54,A56;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 by
XBOOLE_0:def 3;
then Y7 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y7 in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then Y7 in ((X \/ Z1) \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_0:def 3;
then Y7 in ((X \/ Z1) \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD by XBOOLE_0:def 3;
then Y7 in ((X \/ Z1) \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD \/ ZE by XBOOLE_0:def 3;
then Y7 in V by XBOOLE_0:def 3;
hence contradiction by A18,A55,XBOOLE_0:3;
end;
suppose
A57: Y in Z9;
then consider Y1,Y2,Y3,Y4,Y5,Y6 such that
A58: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 & Y5 in Y6 and
A59: Y6 in Y and
A60: Y1 meets X by A3;
Y in union union union union union union union union union X by A3,A57;
then Y6 in union union union union union union union union union union X
by A59,TARSKI:def 4;
then Y6 in ZA by A15,A58,A60;
then Y6 in X \/ Z1 \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/ ZA
by XBOOLE_0:def 3;
then Y6 in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then Y6 in (X \/ Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_0:def 3;
then Y6 in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD by XBOOLE_0:def 3;
then Y6 in (X \/ Z1 \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD \/ ZE by XBOOLE_0:def 3;
then Y6 in V by XBOOLE_0:def 3;
hence contradiction by A18,A59,XBOOLE_0:3;
end;
suppose
A61: Y in ZA;
then consider Y1,Y2,Y3,Y4,Y5 such that
A62: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 & Y4 in Y5 and
A63: Y5 in Y and
A64: Y1 meets X by A15;
Y in union union union union union union union union union union X
by A15,A61;
then Y5 in union union union union union union union union union union
union X by A63,TARSKI:def 4;
then Y5 in ZB by A14,A62,A64;
then Y5 in (X \/ Z1) \/ Z2 \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB by XBOOLE_0:def 3;
then Y5 in (X \/ Z1 \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_0:def 3;
then Y5 in (X \/ Z1 \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD by XBOOLE_0:def 3;
then Y5 in (X \/ Z1 \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD \/ ZE by XBOOLE_0:def 3;
then Y5 in V by XBOOLE_0:def 3;
hence contradiction by A18,A63,XBOOLE_0:3;
end;
suppose
A65: Y in ZB;
then consider Y1,Y2,Y3,Y4 such that
A66: Y1 in Y2 & Y2 in Y3 & Y3 in Y4 and
A67: Y4 in Y and
A68: Y1 meets X by A14;
Y in union union union union union union union union union union (
union X) by A14,A65;
then Y4 in union union union union union union union union union union
union union X by A67,TARSKI:def 4;
then Y4 in ZC by A13,A66,A68;
then Y4 in ((X \/ Z1) \/ Z2) \/ Z3 \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC by XBOOLE_0:def 3;
then Y4 in ((X \/ Z1) \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD by XBOOLE_0:def 3;
then Y4 in ((X \/ Z1) \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD \/ ZE by XBOOLE_0:def 3;
then Y4 in V by XBOOLE_0:def 3;
hence contradiction by A18,A67,XBOOLE_0:3;
end;
suppose
A69: Y in ZC;
then consider Y1,Y2,Y3 such that
A70: Y1 in Y2 & Y2 in Y3 and
A71: Y3 in Y and
A72: Y1 meets X by A13;
Y in union union union union union union union union union union
union union X by A13,A69;
then Y3 in union union union union union union union union union union
union union union X by A71,TARSKI:def 4;
then Y3 in ZD by A12,A70,A72;
then Y3 in ((X \/ Z1) \/ Z2 \/ Z3) \/ Z4 \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD by XBOOLE_0:def 3;
then Y3 in ((X \/ Z1) \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD \/ ZE by XBOOLE_0:def 3;
then Y3 in V by XBOOLE_0:def 3;
hence contradiction by A18,A71,XBOOLE_0:3;
end;
suppose
A73: Y in ZD;
then consider Y1,Y2 such that
A74: Y1 in Y2 and
A75: Y2 in Y and
A76: Y1 meets X by A12;
Y in union union union union union union union union union union
union union union X by A12,A73;
then Y2 in union union union union union union union union union union
union union union union X by A75,TARSKI:def 4;
then Y2 in ZE by A9,A74,A76;
then Y2 in ((X \/ Z1) \/ Z2 \/ Z3 \/ Z4) \/ Z5 \/ Z6 \/ Z7 \/ Z8 \/ Z9 \/
ZA \/ ZB \/ ZC \/ ZD \/ ZE by XBOOLE_0:def 3;
then Y2 in V by XBOOLE_0:def 3;
hence contradiction by A18,A75,XBOOLE_0:3;
end;
suppose
A77: Y in ZE;
then consider Y1 such that
A78: Y1 in Y and
A79: Y1 meets X by A9;
Y in union union union union union union union union union union
union union union union X by A9,A77;
then Y1 in union union union union union union union union union union
union union union union union X by A78,TARSKI:def 4;
then Y1 in ZF by A5,A79;
then Y1 in V by XBOOLE_0:def 3;
hence contradiction by A18,A78,XBOOLE_0:3;
end;
suppose
Y in ZF;
then Y meets X by A5;
hence contradiction by A16,A18,XBOOLE_1:70;
end;
end;
::
:: Tuples for n=9
::
definition
let x1,x2,x3,x4,x5,x6,x7,x8,x9;
func [x1,x2,x3,x4,x5,x6,x7,x8,x9] equals
[[x1,x2,x3,x4,x5,x6,x7,x8],x9];
coherence;
end;
theorem Th161:
[x1,x2,x3,x4,x5,x6,x7,x8,x9] = [[[[[[[[x1,x2],x3],x4],x5],x6], x7],x8],x9]
proof
thus [x1,x2,x3,x4,x5,x6,x7,x8,x9] = [[[[[[[x1,x2,x3],x4],x5],x6],x7],x8],x9]
by MCART_1:def 4
.= [[[[[[[[x1,x2],x3],x4],x5],x6],x7],x8],x9] by MCART_1:def 3;
end;
theorem
[x1,x2,x3,x4,x5,x6,x7,x8,x9] = [[x1,x2,x3,x4,x5,x6,x7],x8,x9] by
MCART_1:def 3;
theorem
[x1,x2,x3,x4,x5,x6,x7,x8,x9] = [[x1,x2,x3,x4,x5,x6],x7,x8,x9] by MCART_1:27;
theorem
[x1,x2,x3,x4,x5,x6,x7,x8,x9] = [[x1,x2,x3,x4,x5],x6,x7,x8,x9] by Th3;
theorem
[x1,x2,x3,x4,x5,x6,x7,x8,x9] = [[x1,x2,x3,x4],x5,x6,x7,x8,x9] by Th41;
theorem
[x1,x2,x3,x4,x5,x6,x7,x8,x9] = [[x1,x2,x3],x4,x5,x6,x7,x8,x9]
proof
thus [x1,x2,x3,x4,x5,x6,x7,x8,x9] = [[[[[[[[x1,x2],x3],x4],x5],x6],x7],x8],
x9] by Th161
.= [[[x1,x2],x3],x4,x5,x6,x7,x8,x9] by Th78
.= [[x1,x2,x3],x4,x5,x6,x7,x8,x9] by MCART_1:def 3;
end;
theorem
[x1,x2,x3,x4,x5,x6,x7,x8,x9] = [[x1,x2],x3,x4,x5,x6,x7,x8,x9] by Th6;
theorem Th168:
[x1,x2,x3,x4,x5,x6,x7,x8,x9] = [y1,y2,y3,y4,y5,y6,y7,y8,y9]
implies x1 = y1 & x2 = y2 & x3 = y3 & x4 = y4 & x5 = y5 & x6 = y6 & x7 = y7 &
x8 = y8 & x9 = y9
proof
assume
A1: [x1,x2,x3,x4,x5,x6,x7,x8,x9] = [y1,y2,y3,y4,y5,y6,y7,y8,y9];
then [x1,x2,x3,x4,x5,x6,x7,x8] = [y1,y2,y3,y4,y5,y6,y7,y8] by ZFMISC_1:27;
hence thesis by A1,Th124,ZFMISC_1:27;
end;
::
:: Cartesian products of nine sets
::
definition
let X1,X2,X3,X4,X5,X6,X7,X8,X9;
func [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] equals
[:[: X1,X2,X3,X4,X5,X6,X7,X8 :],
X9 :];
coherence;
end;
theorem Th169:
[:X1,X2,X3,X4,X5,X6,X7,X8,X9:] = [:[:[:[:[:[:[:[:X1,X2:],X3:],
X4:],X5:],X6:],X7:],X8:],X9:]
proof
thus [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] = [:[:[:[:[:[:[:X1,X2,X3:],X4:],X5:],X6
:],X7:],X8:],X9:]by ZFMISC_1:def 4
.= [:[:[:[:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:],X7:],X8:],X9:] by
ZFMISC_1:def 3;
end;
theorem
[:X1,X2,X3,X4,X5,X6,X7,X8,X9:] = [:[:X1,X2,X3,X4,X5,X6,X7:],X8,X9:] by
ZFMISC_1:def 3;
theorem
[:X1,X2,X3,X4,X5,X6,X7,X8,X9:] = [:[:X1,X2,X3,X4,X5,X6:],X7,X8,X9:] by
MCART_1:49;
theorem
[:X1,X2,X3,X4,X5,X6,X7,X8,X9:] = [:[:X1,X2,X3,X4,X5:],X6,X7,X8,X9:] by Th9;
theorem
[:X1,X2,X3,X4,X5,X6,X7,X8,X9:] = [:[:X1,X2,X3,X4:],X5,X6,X7,X8,X9:] by Th47;
theorem
[:X1,X2,X3,X4,X5,X6,X7,X8,X9:] = [:[:X1,X2,X3:],X4,X5,X6,X7,X8,X9:]
proof
thus [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] = [:[:[:[:[:[:[:[:X1,X2:],X3:],X4:],X5:]
,X6:],X7:],X8:],X9:] by Th169
.= [:[:[:X1,X2:],X3:],X4,X5,X6,X7,X8,X9:] by Th85
.= [:[:X1,X2,X3:],X4,X5,X6,X7,X8,X9:] by ZFMISC_1:def 3;
end;
theorem
[:X1,X2,X3,X4,X5,X6,X7,X8,X9:] = [:[:X1,X2:],X3,X4,X5,X6,X7,X8,X9:] by Th12;
theorem Th176:
X1 <> {} & X2 <> {} & X3 <> {} & X4 <> {} & X5 <> {} & X6 <> {}
& X7 <> {} & X8 <> {} & X9 <> {} iff [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] <> {}
proof
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8<>{}
iff [:X1,X2,X3,X4,X5,X6,X7,X8:] <> {} by Th132;
hence thesis by ZFMISC_1:90;
end;
theorem Th177:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} &
X8<>{} & X9<>{} implies ( [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] = [:Y1,Y2,Y3,Y4,Y5,Y6,
Y7,Y8,Y9:] implies X1=Y1 & X2=Y2 & X3=Y3 & X4=Y4 & X5=Y5 & X6=Y6 & X7=Y7 & X8=
Y8 & X9=Y9)
proof
assume
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <> {};
then
A2: [:X1,X2,X3,X4,X5,X6,X7,X8:] <> {} by Th132;
assume
A3: X9<>{};
assume
A4: [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] = [:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9:];
then [:X1,X2,X3,X4,X5,X6,X7,X8:] = [:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8:] by A2,A3,
ZFMISC_1:110;
hence thesis by A1,A2,A3,A4,Th133,ZFMISC_1:110;
end;
theorem
[:X1,X2,X3,X4,X5,X6,X7,X8,X9:]<>{} & [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] =
[:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9:] implies X1=Y1 & X2=Y2 & X3=Y3 & X4=Y4 & X5=Y5 &
X6=Y6 & X7=Y7 & X8=Y8 & X9=Y9
proof
assume
A1: [:X1,X2,X3,X4,X5,X6,X7,X8,X9:]<>{};
then
A2: X3<>{} & X4<>{} by Th176;
A3: X7<>{} & X8<>{} by A1,Th176;
A4: X5<>{} & X6<>{} by A1,Th176;
A5: X9<>{} by A1,Th176;
X1<>{} & X2<>{} by A1,Th176;
hence thesis by A2,A4,A3,A5,Th177;
end;
theorem
[:X,X,X,X,X,X,X,X,X:] = [:Y,Y,Y,Y,Y,Y,Y,Y,Y:] implies X = Y
proof
assume [:X,X,X,X,X,X,X,X,X:] = [:Y,Y,Y,Y,Y,Y,Y,Y,Y:];
then X<>{} or Y<>{} implies thesis by Th177;
hence thesis;
end;
theorem Th180:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} &
X8<>{} & X9<>{} implies for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:]
ex xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 st x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,
xx9]
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <> {} and
A2: X9<>{};
let x be Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:];
reconsider x9=x as Element of [:[:X1,X2,X3,X4,X5,X6,X7,X8:],X9:];
[:X1,X2,X3,X4,X5,X6,X7,X8:] <> {} by A1,Th132;
then consider
x12345678 being (Element of [:X1,X2,X3,X4,X5,X6,X7,X8:]), xx9 such
that
A3: x9 = [x12345678,xx9] by A2,Lm1;
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8 such that
A4: x12345678 = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8] by A1,Th136;
take xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9;
thus thesis by A3,A4;
end;
definition
let X1,X2,X3,X4,X5,X6,X7,X8,X9;
assume
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <>
{} & X9<>{};
let x be Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:];
func x`1 -> Element of X1 means
:Def37:
x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies it = x1;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 such that
A2: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] by A1,Th180;
take xx1;
thus thesis by A2,Th168;
end;
uniqueness
proof
let y,z be Element of X1;
assume
A3: x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies y = x1;
assume
A4: x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies z = x1;
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 such that
A5: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] by A1,Th180;
y = xx1 by A5,A3;
hence thesis by A5,A4;
end;
func x`2 -> Element of X2 means
:Def38:
x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies it = x2;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 such that
A6: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] by A1,Th180;
take xx2;
thus thesis by A6,Th168;
end;
uniqueness
proof
let y,z be Element of X2;
assume
A7: x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies y = x2;
assume
A8: x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies z = x2;
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 such that
A9: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] by A1,Th180;
y = xx2 by A9,A7;
hence thesis by A9,A8;
end;
func x`3 -> Element of X3 means
:Def39:
x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies it = x3;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 such that
A10: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] by A1,Th180;
take xx3;
thus thesis by A10,Th168;
end;
uniqueness
proof
let y,z be Element of X3;
assume
A11: x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies y = x3;
assume
A12: x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies z = x3;
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 such that
A13: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] by A1,Th180;
y = xx3 by A13,A11;
hence thesis by A13,A12;
end;
func x`4 -> Element of X4 means
:Def40:
x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies it = x4;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 such that
A14: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] by A1,Th180;
take xx4;
thus thesis by A14,Th168;
end;
uniqueness
proof
let y,z be Element of X4;
assume
A15: x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies y = x4;
assume
A16: x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies z = x4;
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 such that
A17: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] by A1,Th180;
y = xx4 by A17,A15;
hence thesis by A17,A16;
end;
func x`5 -> Element of X5 means
:Def41:
x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies it = x5;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 such that
A18: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] by A1,Th180;
take xx5;
thus thesis by A18,Th168;
end;
uniqueness
proof
let y,z be Element of X5;
assume
A19: x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies y = x5;
assume
A20: x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies z = x5;
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 such that
A21: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] by A1,Th180;
y = xx5 by A21,A19;
hence thesis by A21,A20;
end;
func x`6 -> Element of X6 means
:Def42:
x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies it = x6;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 such that
A22: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] by A1,Th180;
take xx6;
thus thesis by A22,Th168;
end;
uniqueness
proof
let y,z be Element of X6;
assume
A23: x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies y = x6;
assume
A24: x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies z = x6;
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 such that
A25: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] by A1,Th180;
y = xx6 by A25,A23;
hence thesis by A25,A24;
end;
func x`7 -> Element of X7 means
:Def43:
x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies it = x7;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 such that
A26: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] by A1,Th180;
take xx7;
thus thesis by A26,Th168;
end;
uniqueness
proof
let y,z be Element of X7;
assume
A27: x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies y = x7;
assume
A28: x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies z = x7;
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 such that
A29: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] by A1,Th180;
y = xx7 by A29,A27;
hence thesis by A29,A28;
end;
func x`8 -> Element of X8 means
:Def44:
x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies it = x8;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 such that
A30: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] by A1,Th180;
take xx8;
thus thesis by A30,Th168;
end;
uniqueness
proof
let y,z be Element of X8;
assume
A31: x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies y = x8;
assume
A32: x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies z = x8;
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 such that
A33: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] by A1,Th180;
y = xx8 by A33,A31;
hence thesis by A33,A32;
end;
func x`9 -> Element of X9 means
:Def45:
x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies it = x9;
existence
proof
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 such that
A34: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] by A1,Th180;
take xx9;
thus thesis by A34,Th168;
end;
uniqueness
proof
let y,z be Element of X9;
assume
A35: x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies y = x9;
assume
A36: x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] implies z = x9;
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 such that
A37: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] by A1,Th180;
y = xx9 by A37,A35;
hence thesis by A37,A36;
end;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8<>{}
& X9 <> {} implies for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] for x1
,x2,x3,x4,x5,x6,x7,x8,x9 st x = [x1,x2,x3,x4,x5,x6,x7,x8,x9] holds x`1 = x1 & x
`2 = x2 & x`3 = x3 & x`4 = x4 & x`5 = x5 & x`6 = x6 & x`7 = x7 & x`8 = x8 & x`9
= x9 by Def37,Def38,Def39,Def40,Def41,Def42,Def43,Def44,Def45;
theorem Th182:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} &
X8<>{} & X9<>{} implies for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:]
holds x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7,x`8,x`9]
proof
assume
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <>
{} & X9<>{};
let x be Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:];
consider xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 such that
A2: x = [xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] by A1,Th180;
thus x = [x`1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] by A1,A2,Def37
.= [x`1,x`2,xx3,xx4,xx5,xx6,xx7,xx8,xx9] by A1,A2,Def38
.= [x`1,x`2,x`3,xx4,xx5,xx6,xx7,xx8,xx9] by A1,A2,Def39
.= [x`1,x`2,x`3,x`4,xx5,xx6,xx7,xx8,xx9] by A1,A2,Def40
.= [x`1,x`2,x`3,x`4,x`5,xx6,xx7,xx8,xx9] by A1,A2,Def41
.= [x`1,x`2,x`3,x`4,x`5,x`6,xx7,xx8,xx9] by A1,A2,Def42
.= [x`1,x`2,x`3,x`4,x`5,x`6,x`7,xx8,xx9] by A1,A2,Def43
.= [x`1,x`2,x`3,x`4,x`5,x`6,x`7,x`8,xx9] by A1,A2,Def44
.= [x`1,x`2,x`3,x`4,x`5,x`6,x`7,x`8,x`9] by A1,A2,Def45;
end;
theorem Th183:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} &
X8<>{} & X9<>{} implies for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:]
holds x`1 = (x qua set)`1`1`1`1`1`1`1`1 & x`2 = (x qua set)`1`1`1`1`1`1`1`2 & x
`3 = (x qua set)`1`1`1`1`1`1`2 & x`4 = (x qua set)`1`1`1`1`1`2 & x`5 = (x qua
set)`1`1`1`1`2 & x`6 = (x qua set)`1`1`1`2 & x`7 = (x qua set)`1`1`2 & x`8 = (x
qua set)`1`2 & x`9 = (x qua set)`2
proof
assume
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <>
{} & X9<>{};
let x be Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:];
thus x`1 = [ x`1, x`2]`1 by MCART_1:7
.= [[x`1, x`2],x`3]`1`1 by MCART_1:7
.= [ x`1, x`2, x`3]`1`1 by MCART_1:def 3
.= [[x`1, x`2, x`3],x`4]`1`1`1 by MCART_1:7
.= [ x`1, x`2, x`3 ,x`4]`1`1`1 by MCART_1:def 4
.= [[x`1, x`2, x`3 ,x`4], x`5]`1`1`1`1 by MCART_1:7
.= [[x`1, x`2, x`3 ,x`4, x`5],x`6]`1`1`1`1`1 by MCART_1:7
.= [[x`1, x`2, x`3 ,x`4, x`5, x`6],x`7]`1`1`1`1`1`1 by MCART_1:7
.= [[x`1, x`2, x`3 ,x`4, x`5, x`6, x`7],x`8]`1`1`1`1`1`1`1 by MCART_1:7
.= [ x`1, x`2, x`3 ,x`4 ,x`5, x`6, x`7, x`8, x`9]`1`1`1`1`1`1`1`1 by
MCART_1:7
.= (x qua set)`1`1`1`1`1`1`1`1 by A1,Th182;
thus x`2 = [ x`1, x`2]`2 by MCART_1:7
.= [[x`1, x`2],x`3]`1`2 by MCART_1:7
.= [ x`1, x`2, x`3]`1`2 by MCART_1:def 3
.= [[x`1, x`2, x`3],x`4]`1`1`2 by MCART_1:7
.= [ x`1, x`2, x`3, x`4]`1`1`2 by MCART_1:def 4
.= [[x`1, x`2, x`3, x`4 ], x`5]`1`1`1`2 by MCART_1:7
.= [[x`1, x`2, x`3, x`4, x`5],x`6]`1`1`1`1`2 by MCART_1:7
.= [[x`1, x`2, x`3, x`4, x`5, x`6],x`7]`1`1`1`1`1`2 by MCART_1:7
.= [[x`1, x`2, x`3, x`4, x`5, x`6, x`7],x`8]`1`1`1`1`1`1`2 by MCART_1:7
.= [ x`1, x`2, x`3, x`4, x`5, x`6, x`7, x`8, x`9]`1`1`1`1`1`1`1`2 by
MCART_1:7
.= (x qua set)`1`1`1`1`1`1`1`2 by A1,Th182;
thus x`3 = [[x`1, x`2],x`3]`2 by MCART_1:7
.= [ x`1, x`2, x`3]`2 by MCART_1:def 3
.= [[x`1, x`2, x`3],x`4]`1`2 by MCART_1:7
.= [ x`1, x`2, x`3, x`4]`1`2 by MCART_1:def 4
.= [[x`1, x`2, x`3, x`4],x`5]`1`1`2 by MCART_1:7
.= [[x`1, x`2, x`3, x`4, x`5],x`6]`1`1`1`2 by MCART_1:7
.= [[x`1, x`2, x`3, x`4, x`5, x`6],x`7]`1`1`1`1`2 by MCART_1:7
.= [[x`1, x`2, x`3, x`4, x`5, x`6, x`7],x`8]`1`1`1`1`1`2 by MCART_1:7
.= [ x`1, x`2, x`3, x`4, x`5, x`6, x`7, x`8, x`9]`1`1`1`1`1`1`2 by
MCART_1:7
.= (x qua set)`1`1`1`1`1`1`2 by A1,Th182;
thus x`4 = [[x`1,x`2,x`3],x`4]`2 by MCART_1:7
.= [ x`1,x`2,x`3, x`4]`2 by MCART_1:def 4
.= [[x`1,x`2,x`3, x`4],x`5]`1`2 by MCART_1:7
.= [[x`1,x`2,x`3, x`4, x`5],x`6]`1`1`2 by MCART_1:7
.= [[x`1,x`2,x`3, x`4, x`5, x`6],x`7]`1`1`1`2 by MCART_1:7
.= [[x`1,x`2,x`3, x`4, x`5, x`6, x`7],x`8]`1`1`1`1`2 by MCART_1:7
.= [ x`1,x`2,x`3, x`4, x`5, x`6, x`7, x`8, x`9]`1`1`1`1`1`2 by MCART_1:7
.= (x qua set)`1`1`1`1`1`2 by A1,Th182;
thus x`5 = [[x`1,x`2,x`3,x`4],x`5]`2 by MCART_1:7
.= [[x`1,x`2,x`3,x`4,x`5],x`6]`1`2 by MCART_1:7
.= [[x`1,x`2,x`3,x`4,x`5,x`6],x`7]`1`1`2 by MCART_1:7
.= [[x`1,x`2,x`3,x`4,x`5,x`6,x`7],x`8]`1`1`1`2 by MCART_1:7
.= [ x`1,x`2,x`3,x`4,x`5,x`6,x`7,x`8, x`9]`1`1`1`1`2 by MCART_1:7
.= (x qua set)`1`1`1`1`2 by A1,Th182;
thus x`6 = [[x`1,x`2,x`3,x`4,x`5],x`6]`2 by MCART_1:7
.= [[x`1,x`2,x`3,x`4,x`5,x`6],x`7]`1`2 by MCART_1:7
.= [[x`1,x`2,x`3,x`4,x`5,x`6,x`7],x`8]`1`1`2 by MCART_1:7
.= [ x`1,x`2,x`3,x`4,x`5,x`6,x`7, x`8, x`9]`1`1`1`2 by MCART_1:7
.= (x qua set)`1`1`1`2 by A1,Th182;
thus x`7 = [[x`1,x`2,x`3,x`4,x`5,x`6],x`7]`2 by MCART_1:7
.= [[x`1,x`2,x`3,x`4,x`5,x`6,x`7],x`8]`1`2 by MCART_1:7
.= [ x`1,x`2,x`3,x`4,x`5,x`6,x`7,x`8, x`9]`1`1`2 by MCART_1:7
.= (x qua set)`1`1`2 by A1,Th182;
thus x`8 = [[x`1,x`2,x`3,x`4,x`5,x`6,x`7],x`8]`2 by MCART_1:7
.= [ x`1,x`2,x`3,x`4,x`5,x`6,x`7,x`8, x`9]`1`2 by MCART_1:7
.= (x qua set)`1`2 by A1,Th182;
thus x`9 = [ x`1,x`2,x`3,x`4,x`5,x`6,x`7,x`8, x`9]`2 by MCART_1:7
.= (x qua set)`2 by A1,Th182;
end;
theorem
[:X1,X2,X3,X4,X5,X6,X7,X8,X9:] meets [:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9:]
implies X1 meets Y1 & X2 meets Y2 & X3 meets Y3 & X4 meets Y4 & X5 meets Y5 &
X6 meets Y6 & X7 meets Y7 & X8 meets Y8 & X9 meets Y9
proof
assume
A1: [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] meets [:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9:];
A2: [:X1,X2,X3,X4,X5,X6,X7,X8:] = [:[:[:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:],
X7:] ,X8:] & [:[:[:[:[:[:[:Y1,Y2:],Y3:],Y4:],Y5:],Y6:],Y7:],Y8:] = [:Y1,Y2,Y3,
Y4,Y5, Y6,Y7,Y8:] by Th126;
[:[:[:[:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:],X7:],X8:],X9:] = [:X1,X2,X3,
X4, X5,X6,X7,X8,X9:] & [:[:[:[:[:[:[:[:Y1,Y2:],Y3:],Y4:],Y5:],Y6:],Y7:],Y8:],Y9
:] = [:Y1,Y2,Y3, Y4,Y5,Y6,Y7,Y8,Y9:] by Th169;
then
[:[:[:[:[:[:[:X1,X2:],X3:],X4:],X5:],X6:],X7:],X8:] meets [:[:[:[:[:[:[:
Y1,Y2:],Y3:],Y4:],Y5:],Y6:],Y7:],Y8:] by A1,ZFMISC_1:104;
hence thesis by A2,A1,Th141,ZFMISC_1:104;
end;
theorem
[:{x1},{x2},{x3},{x4},{x5},{x6},{x7},{x8},{x9}:] = { [x1,x2,x3,x4,x5,
x6,x7,x8,x9] }
proof
thus [:{x1},{x2},{x3},{x4},{x5},{x6},{x7},{x8},{x9}:] = [:[:{x1},{x2}:],{x3}
,{x4},{x5},{x6},{x7},{x8},{x9}:] by Th12
.= [:{[x1,x2]}, {x3},{x4},{x5},{x6},{x7},{x8},{x9}:] by ZFMISC_1:29
.= { [[x1,x2], x3, x4, x5, x6, x7, x8, x9]} by Th142
.= { [x1,x2,x3,x4,x5,x6,x7,x8,x9] } by Th6;
end;
:: 9 - Tuples
reserve x for Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:];
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8<>{}
& X9<>{} implies for x1,x2,x3,x4,x5,x6,x7,x8,x9 st x = [x1,x2,x3,x4,x5,x6,x7,x8
,x9] holds x`1 = x1 & x`2 = x2 & x`3 = x3 & x`4 = x4 & x`5 = x5 & x`6 = x6 & x
`7 = x7 & x`8 = x8 & x`9 = x9 by Def37,Def38,Def39,Def40,Def41,Def42,Def43
,Def44,Def45;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8<>{}
& X9<>{} & (for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 st x = [xx1,xx2,xx3,xx4,xx5
,xx6,xx7,xx8,xx9] holds y1 = xx1) implies y1 =x`1
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <>
{} & X9<>{} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 st x = [xx1,xx2,xx3,xx4,xx5
,xx6,xx7,xx8,xx9] holds y1 = xx1;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7,x`8,x`9] by A1,Th182;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8<>{}
& X9<>{} & (for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 st x = [xx1,xx2,xx3,xx4,xx5
,xx6,xx7,xx8,xx9] holds y2 = xx2) implies y2 =x`2
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <>
{} & X9<>{} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 st x = [xx1,xx2,xx3,xx4,xx5
,xx6,xx7,xx8,xx9] holds y2 = xx2;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7,x`8,x`9] by A1,Th182;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8<>{}
& X9<>{} & (for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 st x = [xx1,xx2,xx3,xx4,xx5
,xx6,xx7,xx8,xx9] holds y3 = xx3) implies y3 =x`3
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <>
{} & X9<>{} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 st x = [xx1,xx2,xx3,xx4,xx5
,xx6,xx7,xx8,xx9] holds y3 = xx3;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7,x`8,x`9] by A1,Th182;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8<>{}
& X9<>{} & (for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 st x = [xx1,xx2,xx3,xx4,xx5
,xx6,xx7,xx8,xx9] holds y4 = xx4) implies y4 =x`4
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <>
{} & X9<>{} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 st x = [xx1,xx2,xx3,xx4,xx5
,xx6,xx7,xx8,xx9] holds y4 = xx4;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7,x`8,x`9] by A1,Th182;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8<>{}
& X9<>{} & (for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 st x = [xx1,xx2,xx3,xx4,xx5
,xx6,xx7,xx8,xx9] holds y5 = xx5) implies y5 =x`5
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <>
{} & X9<>{} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 st x = [xx1,xx2,xx3,xx4,xx5
,xx6,xx7,xx8,xx9] holds y5 = xx5;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7,x`8,x`9] by A1,Th182;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8<>{}
& X9<>{} & (for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 st x = [xx1,xx2,xx3,xx4,xx5
,xx6,xx7,xx8,xx9] holds y6 = xx6) implies y6 =x`6
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <>
{} & X9<>{} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 st x = [xx1,xx2,xx3,xx4,xx5
,xx6,xx7,xx8,xx9] holds y6 = xx6;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7,x`8,x`9] by A1,Th182;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8<>{}
& X9<>{} & (for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 st x = [xx1,xx2,xx3,xx4,xx5
,xx6,xx7,xx8,xx9] holds y7 = xx7) implies y7 =x`7
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <>
{} & X9<>{} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 st x = [xx1,xx2,xx3,xx4,xx5
,xx6,xx7,xx8,xx9] holds y7 = xx7;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7,x`8,x`9] by A1,Th182;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8<>{}
& X9<>{} & (for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 st x = [xx1,xx2,xx3,xx4,xx5
,xx6,xx7,xx8,xx9] holds y8 = xx8) implies y8 =x`8
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <>
{} & X9<>{} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 st x = [xx1,xx2,xx3,xx4,xx5
,xx6,xx7,xx8,xx9] holds y8 = xx8;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7,x`8,x`9] by A1,Th182;
hence thesis by A2;
end;
theorem
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8<>{}
& X9<>{} & (for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 st x = [xx1,xx2,xx3,xx4,xx5
,xx6,xx7,xx8,xx9] holds y9 = xx9) implies y9 =x`9
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <>
{} & X9<>{} and
A2: for xx1,xx2,xx3,xx4,xx5,xx6,xx7,xx8,xx9 st x = [xx1,xx2,xx3,xx4,xx5
,xx6,xx7,xx8,xx9] holds y9 = xx9;
x = [x`1,x`2,x`3,x`4,x`5,x`6,x`7,x`8,x`9] by A1,Th182;
hence thesis by A2;
end;
theorem
y in [: X1,X2,X3,X4,X5,X6,X7,X8,X9 :] implies ex x1,x2,x3,x4,x5,x6,x7,
x8,x9 st x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7
in X7 & x8 in X8 & x9 in X9 & y = [x1,x2,x3,x4,x5,x6,x7,x8,x9]
proof
assume y in [: X1,X2,X3,X4,X5,X6,X7,X8,X9 :];
then consider x12345678, x9 being set such that
A1: x12345678 in [:X1,X2,X3,X4,X5,X6,X7,X8:] and
A2: x9 in X9 and
A3: y = [x12345678,x9] by ZFMISC_1:def 2;
consider x1, x2, x3, x4, x5, x6, x7, x8 such that
A4: x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7
in X7 & x8 in X8 and
A5: x12345678 = [x1,x2,x3,x4,x5,x6,x7,x8] by A1,Th152;
y = [x1,x2,x3,x4,x5,x6,x7,x8,x9] by A3,A5;
hence thesis by A2,A4;
end;
theorem
[x1,x2,x3,x4,x5,x6,x7,x8,x9] in [: X1,X2,X3,X4,X5,X6,X7,X8,X9 :] iff
x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7 in X7 & x8
in X8 & x9 in X9
proof
A1: [x1,x2] in [:X1,X2:] iff x1 in X1 & x2 in X2 by ZFMISC_1:87;
[:X1,X2,X3,X4,X5,X6,X7,X8,X9:] = [:[:X1,X2:],X3,X4,X5,X6,X7,X8,X9:] & [
x1,x2,x3,x4,x5,x6,x7,x8,x9] = [[x1,x2],x3,x4,x5,x6,x7,x8,x9] by Th6,Th12;
hence thesis by A1,Th153;
end;
theorem
(for y holds y in Z iff ex x1,x2,x3,x4,x5,x6,x7,x8,x9 st x1 in X1 & x2
in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7 in X7 & x8 in X8 & x9 in
X9 & y = [x1,x2,x3,x4,x5,x6,x7,x8,x9]) implies Z = [: X1,X2,X3,X4,X5,X6,X7,X8,
X9 :]
proof
assume
A1: for y holds y in Z iff ex x1,x2,x3,x4,x5,x6,x7,x8,x9 st x1 in X1 &
x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 & x7 in X7 & x8 in X8 & x9
in X9 & y = [x1,x2,x3,x4,x5,x6,x7,x8,x9];
now
let y;
thus y in Z implies y in [:[:X1,X2,X3,X4,X5,X6,X7,X8:],X9:]
proof
assume y in Z;
then consider x1,x2,x3,x4,x5,x6,x7,x8,x9 such that
A2: x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6
& x7 in X7 & x8 in X8 and
A3: x9 in X9 & y = [x1,x2,x3,x4,x5,x6,x7,x8,x9] by A1;
[x1,x2,x3,x4,x5,x6,x7,x8] in [:X1,X2,X3,X4,X5,X6,X7,X8:] by A2,Th153;
hence thesis by A3,ZFMISC_1:def 2;
end;
assume y in [:[:X1,X2,X3,X4,X5,X6,X7,X8:],X9:];
then consider x12345678,x9 being set such that
A4: x12345678 in [:X1,X2,X3,X4,X5,X6,X7,X8:] and
A5: x9 in X9 and
A6: y = [x12345678,x9] by ZFMISC_1:def 2;
consider x1,x2,x3,x4,x5,x6,x7,x8 such that
A7: x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & x5 in X5 & x6 in X6 &
x7 in X7 & x8 in X8 and
A8: x12345678 = [x1,x2,x3,x4,x5,x6,x7,x8] by A4,Th152;
y = [x1,x2,x3,x4,x5,x6,x7,x8,x9] by A6,A8;
hence y in Z by A1,A5,A7;
end;
hence thesis by TARSKI:1;
end;
theorem Th199:
X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} &
X8<>{} & X9<>{} & Y1<>{} & Y2<>{} & Y3<>{} & Y4<>{} & Y5<>{} & Y6<>{} & Y7<>{}
& Y8<>{} & Y9<>{} implies for x being (Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9
:]), y being Element of [:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9:] holds x = y implies x`1
= y`1 & x`2 = y`2 & x`3 = y`3 & x`4 = y`4 & x`5 = y`5 & x`6 = y`6 & x`7 = y`7 &
x`8 = y`8 & x`9 = y`9
proof
assume that
A1: X1<>{} & X2<>{} & X3<>{} & X4<>{} & X5<>{} & X6<>{} & X7<>{} & X8 <>
{} & X9<>{} and
A2: Y1<>{} & Y2<>{} & Y3<>{} & Y4<>{} & Y5<>{} & Y6<>{} & Y7<>{} & Y8
<>{} & Y9<>{};
let x be Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:];
let y be Element of [:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9:];
assume
A3: x = y;
thus x`1 = (x qua set)`1`1`1`1`1`1`1`1 by A1,Th183
.= y`1 by A2,A3,Th183;
thus x`2 = (x qua set)`1`1`1`1`1`1`1`2 by A1,Th183
.= y`2 by A2,A3,Th183;
thus x`3 = (x qua set)`1`1`1`1`1`1`2 by A1,Th183
.= y`3 by A2,A3,Th183;
thus x`4 = (x qua set)`1`1`1`1`1`2 by A1,Th183
.= y`4 by A2,A3,Th183;
thus x`5 = (x qua set)`1`1`1`1`2 by A1,Th183
.= y`5 by A2,A3,Th183;
thus x`6 = (x qua set)`1`1`1`2 by A1,Th183
.= y`6 by A2,A3,Th183;
thus x`7 = (x qua set)`1`1`2 by A1,Th183
.= y`7 by A2,A3,Th183;
thus x`8 = (x qua set)`1`2 by A1,Th183
.= y`8 by A2,A3,Th183;
thus x`9 = (x qua set)`2 by A1,Th183
.= y`9 by A2,A3,Th183;
end;
theorem
for x being Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] st x in [:A1,A2,
A3,A4,A5,A6,A7,A8,A9:] holds x`1 in A1 & x`2 in A2 & x`3 in A3 & x`4 in A4 & x
`5 in A5 & x`6 in A6 & x`7 in A7 & x`8 in A8 & x`9 in A9
proof
let x be Element of [:X1,X2,X3,X4,X5,X6,X7,X8,X9:];
assume
A1: x in [:A1,A2,A3,A4,A5,A6,A7,A8,A9:];
then reconsider y = x as Element of [:A1,A2,A3,A4,A5,A6,A7,A8,A9:];
A2<>{} by A1,Th176;
then
A2: y`2 in A2;
A8<> {} by A1,Th176;
then
A3: y`8 in A8;
A7<>{} by A1,Th176;
then
A4: y`7 in A7;
A6<>{} by A1,Th176;
then
A5: y`6 in A6;
A5<>{} by A1,Th176;
then
A6: y`5 in A5;
A9<>{} by A1,Th176;
then
A7: y`9 in A9;
A4<>{} by A1,Th176;
then
A8: y`4 in A4;
A3<>{} by A1,Th176;
then
A9: y`3 in A3;
A1<>{} by A1,Th176;
then y`1 in A1;
hence thesis by A2,A9,A8,A6,A5,A4,A3,A7,Th199;
end;
theorem Th201:
X1 c= Y1 & X2 c= Y2 & X3 c= Y3 & X4 c= Y4 & X5 c= Y5 & X6 c= Y6
& X7 c= Y7 & X8 c= Y8 & X9 c= Y9 implies [:X1,X2,X3,X4,X5,X6,X7,X8,X9:] c= [:Y1
,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9:]
proof
assume X1 c= Y1 & X2 c= Y2 & X3 c= Y3 & X4 c= Y4 & X5 c= Y5 & X6 c= Y6 & X7
c= Y7 & X8 c= Y8;
then
A1: [:X1,X2,X3,X4,X5,X6,X7,X8:] c= [:Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8:] by Th157;
assume X9 c= Y9;
hence thesis by A1,ZFMISC_1:96;
end;
theorem
[:A1,A2,A3,A4,A5,A6,A7,A8,A9:] is Subset of [:X1,X2,X3,X4,X5,X6,X7,X8,
X9:] by Th201;