(** 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 empty_b : mem y empty = false.