proposition: Lemma HF_numeral_trans T n x y : HFeq <<= T -> T ⊢ x ∈ tnumeral n → y ∈ x → y ∈ tnumeral n.
proof: Proof. intros HT. induction n; cbn. - apply II, Exp. eapply IE. apply HF_eset'. all: auto. - apply bunion_use; trivial. + rewrite !imps. rewrite !imps in IHn. apply HF_bunion_el1; auto. + apply II. apply HF_bunion_el'. auto. apply DI1. eapply HF_eq_elem; auto. apply HF_refl'. auto. Qed.
imports: Undecidability.FOL.ZF Undecidability.FOL.Syntax.Facts Undecidability.FOL.Semantics.Tarski.FullFacts Undecidability.FOL.Sets.ZF Undecidability.FOL.Reductions.PCPb_to_HF Undecidability.PCP(PCP Util.PCP_facts Reductions.PCPb_iff_dPCPb) Undecidability(Shared.ListAutomation)