Hint Resolve set_union_intro2 set_union_intro1.

(* Specifications: *)
Lemma set_union_intro :
forall (a : A) (x y : set),
set_In a x \/ set_In a y -> set_In a (set_union x y).
Lemma set_union_elim :
forall (a : A) (x y : set),
set_In a (set_union x y) -> set_In a x \/ set_In a y.
Lemma set_inter_intro :
forall (a : A) (x y : set),
set_In a x -> set_In a y -> set_In a (set_inter x y).
Lemma set_inter_elim1 :
forall (a : A) (x y : set), set_In a (set_inter x y) -> set_In a x.
Lemma set_inter_elim2 :
forall (a : A) (x y : set), set_In a (set_inter x y) -> set_In a y.