let X1, X2 be set ; :: thesis: ( ( for x being object holds
( x in X1 iff ex Y being set st
( x in Y & Y in X ) ) ) & ( for x being object holds
( x in X2 iff ex Y being set st
( x in Y & Y in X ) ) ) implies X1 = X2 )
assume that
A1: for x being object holds
( x in X1 iff ex Y being set st
( x in Y & Y in X ) ) and
A2: for x being object holds
( x in X2 iff ex Y being set st
( x in Y & Y in X ) ) ; :: thesis: X1 = X2

now :: thesis: for x being object holds
( x in X1 iff x in X2 )

let x be object ; :: thesis: ( x in X1 iff x in X2 )
( x in X1 iff ex Y being set st
( x in Y & Y in X ) ) by A1;
hence ( x in X1 iff x in X2 ) by A2; :: thesis: verum

end;

hence X1 = X2 by TARSKI_0:2; :: thesis: verum