proposition: Lemma opair_inj1 T x y x' y' : ZFeq' <<= T -> T ⊢ opair x y ≡ opair x' y' -> T ⊢ x ≡ x'.
proof: Proof. intros HT He. assert (H : T ⊢ {x; x} ∈ opair x y). { apply ZF_pair_el'; trivial. apply DI1. now apply ZF_refl'. } eapply ZF_eq_elem in H; try apply He; try apply ZF_refl'; trivial. apply ZF_pair_el' in H; trivial. apply (DE H); eapply sing_pair1; try apply prv_T1; 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)