let P, Q be preBoolean set ; :: thesis: ( ( for X being set holds
( X in P iff ( X c= A & X is finite ) ) ) & ( for X being set holds
( X in Q iff ( X c= A & X is finite ) ) ) implies P = Q )
assume that
A11: for X being set holds
( X in P iff ( X c= A & X is finite ) ) and
A12: for X being set holds
( X in Q iff ( X c= A & X is finite ) ) ; :: thesis: P = Q
for x being object holds
( x in P iff x in Q )

proof

let x be object ; :: thesis: ( x in P iff x in Q )
reconsider xx = x as set by TARSKI:1;
thus ( x in P implies x in Q ) :: thesis: ( x in Q implies x in P )

proof

assume x in P ; :: thesis: x in Q
then ( xx c= A & xx is finite ) by A11;
hence x in Q by A12; :: thesis: verum

end;

thus ( x in Q implies x in P ) :: thesis: verum

proof

assume x in Q ; :: thesis: x in P
then ( xx c= A & xx is finite ) by A12;
hence x in P by A11; :: thesis: verum

end;

end;

hence P = Q by TARSKI:2; :: thesis: verum