Lemma tightapSet_dec {X : tightapSet} : LEM → ∏ x y : X, (x != y <-> x ≠ y). Proof. intros Hdec x y. destruct (Hdec (x ≠ y)) as [ Hneq | Heq ]. - split. + intros _ ; apply Hneq. + intros _ Heq. rewrite <- Heq in Hneq. revert Hneq. now apply isirrefltightapSet. - split. + intros Hneq. apply fromempty, Hneq. now apply istighttightapSet. + intros Hneq. exact (fromempty (Heq Hneq)). Qed.
Require Import UniMath.Foundations.Propositions. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.DecidablePropositions.