(** Useful tactic for simplifying expressions like [In y (add x (union s s'))] *)

Ltac set_iff :=
 repeat (progress (
  rewrite add_iff || rewrite remove_iff || rewrite singleton_iff
  || rewrite union_iff || rewrite inter_iff || rewrite diff_iff
  || rewrite empty_iff)).

Lemma remove_neq_b : ~ E.eq x y -> mem y (remove x s) = mem y s.