let Z, X1, X2, X3 be set ; :: thesis: ( ( for z being set holds
( z in Z iff ex x1, x2, x3 being set st
( x1 in X1 & x2 in X2 & x3 in X3 & z = [x1,x2,x3] ) ) ) implies Z = [:X1,X2,X3:] )
assume A1: for z being set holds
( z in Z iff ex x1, x2, x3 being set st
( x1 in X1 & x2 in X2 & x3 in X3 & z = [x1,x2,x3] ) ) ; :: thesis: Z = [:X1,X2,X3:]

now

let z be set ; :: thesis: ( ( z in Z implies z in [:[:X1,X2:],X3:] ) & ( z in [:[:X1,X2:],X3:] implies z in Z ) )
thus ( z in Z implies z in [:[:X1,X2:],X3:] ) :: thesis: ( z in [:[:X1,X2:],X3:] implies z in Z )

proof

assume z in Z ; :: thesis: z in [:[:X1,X2:],X3:]
then consider x1, x2, x3 being set such that
A2: ( x1 in X1 & x2 in X2 & x3 in X3 & z = [x1,x2,x3] ) by A1;
( [x1,x2] in [:X1,X2:] & x3 in X3 ) by A2, ZFMISC_1:def 2;
hence z in [:[:X1,X2:],X3:] by A2, ZFMISC_1:def 2; :: thesis: verum

end;

assume z in [:[:X1,X2:],X3:] ; :: thesis: z in Z
then consider x12, x3 being set such that
A3: x12 in [:X1,X2:] and
A4: x3 in X3 and
A5: z = [x12,x3] by ZFMISC_1:def 2;
consider x1, x2 being set such that
A6: x1 in X1 and
A7: x2 in X2 and
A8: x12 = [x1,x2] by A3, ZFMISC_1:def 2;
z = [x1,x2,x3] by A5, A8;
hence z in Z by A1, A4, A6, A7; :: thesis: verum

end;

then Z = [:[:X1,X2:],X3:] by TARSKI:2;
hence Z = [:X1,X2,X3:] by ZFMISC_1:def 3; :: thesis: verum