let X1, X2 be Subset of F₁(); :: thesis: ( ( for y being set holds
( y in X1 iff P₁[y] ) ) & ( for y being set holds
( y in X2 iff P₁[y] ) ) implies X1 = X2 )
assume that
A1: for y being set holds
( y in X1 iff P₁[y] ) and
A2: for y being set holds
( y in X2 iff P₁[y] ) ; :: thesis: X1 = X2
for x being set holds
( x in X1 iff x in X2 )

proof

let x be set ; :: thesis: ( x in X1 iff x in X2 )

hereby :: thesis: ( x in X2 implies x in X1 )

assume x in X1 ; :: thesis: x in X2
then P₁[x] by A1;
hence x in X2 by A2; :: thesis: verum

end;

assume x in X2 ; :: thesis: x in X1
then P₁[x] by A2;
hence x in X1 by A1; :: thesis: verum

end;

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