let D be set ; :: thesis: for F, G being Subset-Family of D st ( for P being Subset of D holds
( P in F iff P in G ) ) holds
F = G
let F, G be Subset-Family of D; :: thesis: ( ( for P being Subset of D holds
( P in F iff P in G ) ) implies F = G )
assume A1: for P being Subset of D holds
( P in F iff P in G ) ; :: thesis: F = G
thus F c= G :: according to XBOOLE_0:def 10 :: thesis: G c= F

proof

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in F or x in G )
assume x in F ; :: thesis: x in G
hence x in G by A1; :: thesis: verum

end;

let x be set ; :: according to TARSKI:def 3 :: thesis: ( not x in G or x in F )
assume x in G ; :: thesis: x in F
hence x in F by A1; :: thesis: verum