Proof with auto.
intro A; rewrite elements_In in A.
solve [apply setd_add_In_inv in A; exact A].
Qed.

(* Specifications: *)
Lemma setd_rem_In_inv : M.In y (M.remove x s) -> M.In y s.
Proof with auto. rewrite M.remove_spec; intros [A _]... Qed.
Lemma setd_rem_In_inv_elems :
In y (M.elements (M.remove x s)) -> In y (M.elements s).