proposition: Lemma ZF_pair_el' T x y z : ZFeq' <<= T -> T ⊢ (z ≡ x ∨ z ≡ y) <-> T ⊢ z ∈ {x; y}.
proof: Proof. intros HT; split; intros H; eapply IE; try apply H. all: assert (HP : T ⊢ ax_pair) by (apply Ctx; firstorder). all: apply (AllE y), (AllE x), (AllE z) in HP; cbn in HP; subsimpl_in HP. - eapply CE2, HP. - eapply CE1, HP. 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)