Proposition irrefl_is_hausdorff_separated (x : X) : ¬(is_hausdorff_separated x x). Proof. use factor_through_squash. { apply isapropempty. } intros H. induction H as [ P₁ [ P₂ [ Px [ Px' H ]]]]. exact (H x (Px ,, Px')). Qed.
Require Import UniMath.MoreFoundations.All. Require Import UniMath.OrderTheory.DCPOs.Core.DirectedSets. Require Import UniMath.OrderTheory.DCPOs.Core.Basics. Require Import UniMath.OrderTheory.DCPOs.Core.WayBelow. Require Import UniMath.OrderTheory.DCPOs.Core.ScottTopology. Require Import UniMath.OrderTheory.DCPOs.Core.IntrinsicApartness. Require Import UniMath.OrderTheory.DCPOs.Basis.Continuous. Require Import UniMath.OrderTheory.DCPOs.Basis.Basis. Require Import UniMath.OrderTheory.DCPOs.Elements.Sharp.