Lemma subset_lattice_is_bounded : bounded_latticeop subset_lattice (emptysubtype X) (totalsubtype X). Proof. use make_dirprod. - intros x; cbn. apply (invweq (hsubtype_univalence _ _)), subtype_equal_cond; use make_dirprod. + intros ? in_union. refine (hinhuniv _ in_union); clear in_union; intro in_union. induction in_union as [b bin]; induction b; cbn in bin. * induction bin. * assumption. + intros ? ?; apply hinhpr; exists false; assumption. - intros x; cbn. apply (invweq (hsubtype_univalence _ _)), subtype_equal_cond; use make_dirprod. + intros ? in_intersection; exact (in_intersection false). + intros ? ? b; induction b; cbn. * exact tt. * assumption. Qed.
Require Import UniMath.Foundations.Preamble. Require Import UniMath.Foundations.PartA. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Sets. Require Import UniMath.Algebra.BinaryOperations. Require Import UniMath.OrderTheory.Lattice.Lattice. Require Import UniMath.OrderTheory.Lattice.Bounded. Require Import UniMath.OrderTheory.Lattice.Distributive. Require Import UniMath.OrderTheory.Lattice.Complement. Require Import UniMath.OrderTheory.Lattice.Boolean.