let z be object ; :: thesis: for X1, X2, X3, X4 being set st z in [:X1,X2,X3,X4:] holds
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] )
let X1, X2, X3, X4 be set ; :: thesis: ( z in [:X1,X2,X3,X4:] implies 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] ) )
assume z in [:X1,X2,X3,X4:] ; :: thesis: 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] )
then z in [:[:X1,X2,X3:],X4:] by ZFMISC_1:def 4;
then consider x123, x4 being object such that
A1: x123 in [:X1,X2,X3:] and
A2: x4 in X4 and
A3: z = [x123,x4] by ZFMISC_1:def 2;
consider x1, x2, x3 being object such that
A4: ( x1 in X1 & x2 in X2 & x3 in X3 ) and
A5: x123 = [x1,x2,x3] by A1, Th54;
z = [x1,x2,x3,x4] by A3, A5;
hence 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] ) by A2, A4; :: thesis: verum