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

now

let x be set ; :: thesis: ( x in X1 iff x in X2 )
( x in X1 iff P₁[x] ) by A1;
hence ( x in X1 iff x in X2 ) by A2; :: thesis: verum

end;

hence X1 = X2 by TARSKI:1; :: thesis: verum