Proposition is_hausdorff_separated_not_le {x y : X} (H : is_hausdorff_separated x y) : ¬(x ⊑ y). Proof. intro p. revert H. use factor_through_squash. { apply isapropempty. } intros H. induction H as [ P₁ [ P₂ [ Px [ Py H ]]]]. refine (H y (_ ,, _)). - refine (is_scott_open_upper_set P₁ Px p). - exact Py. 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.