proposition: Lemma ZF_numeral_trans T n x y : ZFeq' <<= T -> T ⊢ x ∈ tnumeral n → y ∈ x → y ∈ tnumeral n.
proof: Proof. intros HT. induction n; cbn. - apply II, Exp. eapply IE. apply ZF_eset'. all: auto. - apply bunion_use; trivial. + rewrite !imps. rewrite !imps in IHn. apply ZF_bunion_el1; auto. + apply II. apply ZF_bunion_el'. auto. apply DI1. eapply ZF_eq_elem; auto. apply ZF_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_ZF Undecidability.PCP(PCP Util.PCP_facts Reductions.PCPb_iff_dPCPb) Undecidability(Shared.ListAutomation)