let X, Y be set ; :: thesis: ( [:X,X:] = [:Y,Y:] implies X = Y )
assume A1: [:X,X:] = [:Y,Y:] ; :: thesis: X = Y
for x being object holds
( x in X iff x in Y )

proof

let x be object ; :: thesis: ( x in X iff x in Y )
( x in X iff [x,x] in [:X,X:] ) by Lm16;
hence ( x in X iff x in Y ) by A1, Lm16; :: thesis: verum

end;

hence X = Y by TARSKI:2; :: thesis: verum