proposition: Lemma WO_AC {LEM : ExcludedMiddle} : (forall (X : HSet), hexists (fun (A : Ordinal) => InjectsInto X A)) -> Choice_type.
proof: Proof. intros H X Y R HR. specialize (H Y). eapply merely_destruct; try apply H. intros [A HA]. eapply merely_destruct; try apply HA. intros [f Hf]. apply tr. unshelve eexists. - intros x. assert (HR' : hexists (fun y => merely (R x y * forall y', R x y' -> f y < f y' \/ f y = f y'))). + pose proof (HAR := ordinal_has_minimal_hsolutions A (fun a => Build_HProp (merely (exists y, f y = a /\ R x y)))). eapply merely_destruct; try apply HAR. * eapply merely_destruct; try apply (HR x). intros [y Hy]. apply tr. exists (f y). apply tr. exists y. now split. * intros [a [H1 H2]]. eapply merely_destruct; try apply H1. intros [y [<- Hy]]. apply tr. exists y. apply tr. split; trivial. intros y' Hy'. apply H2. apply tr. exists y'. split; trivial. + edestruct (@iota Y) as [y Hy]; try exact y. 2: split; try apply HR'. 1: exact _. intros y y' Hy Hy'. eapply merely_destruct; try apply Hy. intros [H1 H2]. eapply merely_destruct; try apply Hy'. intros [H3 H4]. apply Hf. eapply merely_destruct; try apply (H2 y'); trivial. intros [H5|H5]; trivial. eapply merely_destruct; try apply (H4 y); trivial. intros [H6| -> ]; trivial. apply Empty_rec. apply (irreflexive_ordinal_relation _ _ _ (f y)). apply (ordinal_transitivity _ (f y')); trivial. - intros x. cbn. destruct iota as [y Hy]. eapply merely_destruct; try apply Hy. now intros []. Qed.
imports: HoTT(ExcludedMiddle abstract_algebra) HoTT(HIT.unique_choice) HoTT(Spaces.Card) HoTT.Sets(Ordinals)