let X1, X2, X3, X4, Z be set ; ( ( for z being object holds
( z in Z iff ex x1, x2, x3, x4 being object st
( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & z = [x1,x2,x3,x4] ) ) ) implies Z = [:X1,X2,X3,X4:] )
assume A1:
for z being object holds
( z in Z iff ex x1, x2, x3, x4 being object st
( x1 in X1 & x2 in X2 & x3 in X3 & x4 in X4 & z = [x1,x2,x3,x4] ) )
; Z = [:X1,X2,X3,X4:]
now for z being object holds
( ( z in Z implies z in [:[:X1,X2,X3:],X4:] ) & ( z in [:[:X1,X2,X3:],X4:] implies z in Z ) )let z be
object ;
( ( z in Z implies z in [:[:X1,X2,X3:],X4:] ) & ( z in [:[:X1,X2,X3:],X4:] implies z in Z ) )thus
(
z in Z implies
z in [:[:X1,X2,X3:],X4:] )
( z in [:[:X1,X2,X3:],X4:] implies z in Z )proof
assume
z in Z
;
z in [:[:X1,X2,X3:],X4:]
then consider x1,
x2,
x3,
x4 being
object such that A2:
(
x1 in X1 &
x2 in X2 &
x3 in X3 )
and A3:
(
x4 in X4 &
z = [x1,x2,x3,x4] )
by A1;
[x1,x2,x3] in [:X1,X2,X3:]
by A2, Th55;
hence
z in [:[:X1,X2,X3:],X4:]
by A3, ZFMISC_1:def 2;
verum
end; assume
z in [:[:X1,X2,X3:],X4:]
;
z in Zthen consider x123,
x4 being
object such that A4:
x123 in [:X1,X2,X3:]
and A5:
x4 in X4
and A6:
z = [x123,x4]
by ZFMISC_1:def 2;
consider x1,
x2,
x3 being
object such that A7:
(
x1 in X1 &
x2 in X2 &
x3 in X3 )
and A8:
x123 = [x1,x2,x3]
by A4, Th54;
z = [x1,x2,x3,x4]
by A6, A8;
hence
z in Z
by A1, A5, A7;
verum end;
then
Z = [:[:X1,X2,X3:],X4:]
by TARSKI:2;
hence
Z = [:X1,X2,X3,X4:]
by ZFMISC_1:def 4; verum