proposition: Lemma HF_sub_pair T x y x' y' : HFeq <<= T -> T ⊢ x ≡ x' -> T ⊢ y ≡ y' -> T ⊢ {x; y} ⊆ {x'; y'}.
proof: Proof. intros HT H1 H2. prv_all z. apply II. apply HF_pair_el'; auto. eapply DE. - apply HF_pair_el'; auto. - apply DI1. eapply HF_trans'; auto. eapply Weak; eauto. - apply DI2. eapply HF_trans'; auto. eapply Weak; eauto. 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)