proposition: Lemma bZF_elem { p : peirce } T x x' y y' : binZF <<= T -> T ⊢ x ≡' x' -> T ⊢ y ≡' y' -> T ⊢ x ∈' y -> T ⊢ x' ∈' y'.
proof: Proof. intros H1 H2 H3 H4. eapply IE; try apply H4. eapply IE; try apply H3. eapply IE; try apply H2. assert (H : T ⊢ ax_eq_elem') by (apply Ctx; firstorder). apply (AllE x), (AllE y), (AllE x'), (AllE y') in H; cbn in H. unfold eq' in H. cbn in H. subsimpl_in H. apply H. Qed.
imports: Undecidability.FOL.Syntax.Facts Undecidability.FOL.Semantics.Tarski.FullFacts Undecidability.FOL.Semantics.Tarski.FullSoundness Undecidability.FOL.Deduction.FullNDFacts Undecidability.FOL.Reductions.PCPb_to_binZF Undecidability.FOL.ZF Undecidability.FOL.FST Undecidability.FOL.Sets(Models.FST_model Models.ZF_model binFST binZF) Lia Undecidability(Shared.ListAutomation)