let X1, X2 be set ; :: thesis: ( ( for u being set holds
( u in X1 iff ( u <> 0. V & u is Element of V ) ) ) & ( for u being set holds
( u in X2 iff ( u <> 0. V & u is Element of V ) ) ) implies X1 = X2 )
assume that
A2: for u being set holds
( u in X1 iff ( u <> 0. V & u is Element of V ) ) and
A3: for u being set holds
( u in X2 iff ( u <> 0. V & u is Element of V ) ) ; :: thesis: X1 = X2
for u being set holds
( u in X1 iff u in X2 )

proof

let u be set ; :: thesis: ( u in X1 iff u in X2 )
thus ( u in X1 implies u in X2 ) :: thesis: ( u in X2 implies u in X1 )

proof

assume u in X1 ; :: thesis: u in X2
then ( u <> 0. V & u is Element of V ) by A2;
hence u in X2 by A3; :: thesis: verum

end;

thus ( u in X2 implies u in X1 ) :: thesis: verum

proof

assume u in X2 ; :: thesis: u in X1
then ( u <> 0. V & u is Element of V ) by A3;
hence u in X1 by A2; :: thesis: verum

end;

end;

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