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 object holds
( x in X1 iff x in X2 ) by A1, A2;
hence X1 = X2 by TARSKI:2; :: thesis: verum