Definition least_upper_bound_univ {I} {p : I -> P} (x : least_upper_bound p) (x' : P) : x ≤ x' <-> isupperbound p x'. Proof. split. - intros xx' i. eapply istrans_posetRelation; try eassumption. apply least_upper_bound_is_upper_bound. - intros H. refine (least_is_least _ x (_,,H)). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.OrderTheory.OrderedSets.WellOrderedSets. Require Import UniMath.OrderTheory.DCPOs.AlternativeDefinitions.Dcpo. Require Import UniMath.OrderTheory.Posets.Subposets.