let z be object ; for X1, X2, X3 being set st z in [:X1,X2,X3:] holds
ex x1, x2, x3 being object st
( x1 in X1 & x2 in X2 & x3 in X3 & z = [x1,x2,x3] )
let X1, X2, X3 be set ; ( z in [:X1,X2,X3:] implies ex x1, x2, x3 being object st
( x1 in X1 & x2 in X2 & x3 in X3 & z = [x1,x2,x3] ) )
assume
z in [:X1,X2,X3:]
; ex x1, x2, x3 being object st
( x1 in X1 & x2 in X2 & x3 in X3 & z = [x1,x2,x3] )
then
z in [:[:X1,X2:],X3:]
by ZFMISC_1:def 3;
then consider x12, x3 being object such that
A1:
x12 in [:X1,X2:]
and
A2:
x3 in X3
and
A3:
z = [x12,x3]
by ZFMISC_1:def 2;
consider x1, x2 being object such that
A4:
( x1 in X1 & x2 in X2 )
and
A5:
x12 = [x1,x2]
by A1, ZFMISC_1:def 2;
z = [x1,x2,x3]
by A3, A5;
hence
ex x1, x2, x3 being object st
( x1 in X1 & x2 in X2 & x3 in X3 & z = [x1,x2,x3] )
by A2, A4; verum