Hint Resolve set_inter_elim1 set_inter_elim2.

(* Specifications: *)
Lemma set_inter_elim :
forall (a : A) (x y : set),
set_In a (set_inter x y) -> set_In a x /\ set_In a y.
Lemma set_diff_intro :
forall (a : A) (x y : set),
set_In a x -> ~ set_In a y -> set_In a (set_diff x y).
Lemma set_diff_elim1 :
forall (a : A) (x y : set), set_In a (set_diff x y) -> set_In a x.