proposition: Lemma set0_P x: empty x <-> x = emptyset.
proof: Proof. split; last by move => -> y /in_set0. by move => Iyx; set_extens t => Ht; [ case: (Iyx t) | case: (in_set0 Ht) ]. Qed.
imports: mathcomp(ssreflect ssrfun ssrbool eqtype ssrnat)