Commit f75bc28d by Samuel Ben Hamou

### Elucubrations sur les singletons.

parent 3d63606d
 ... ... @@ -243,8 +243,42 @@ Proof. apply R_Ax. apply in_eq. Qed. (* utiliser l'axiome de la paire et défaire puis refaire quantificateurs OU paire -> existence de {x,x}; skolem; séparation -> {x} = {x,x} *) Lemma singleton : IsTheorem Intuiti ZF (∀∃∀ (#0 ∈ #1 <-> #0 = #2)). Admitted. Proof. thm. exists [pairing]. split; auto. - simpl. rewrite Forall_forall. intros. destruct H. + rewrite<- H. unfold IsAx. left. compute; intuition. + inversion H. - apply R_All_i with (x := "x"); [ calc | cbn ]. apply R_Ex_i with (t := FVar "s"); cbn. apply R_All_i with (x := "y"); [ calc | cbn ]. assert (Pr Intuiti ([pairing] ⊢ pairing)). { apply R_Ax. apply in_eq. } unfold pairing in H. apply R_All_e with (t := FVar "x") in H; cbn in H. apply R_All_e with (t := FVar "x") in H; cbn in H. (* il se passe un truc ici *) apply R_And_i. + apply R_Imp_i. apply R_Imp_e with (A := FVar "y" ∈ FVar "s" \/ FVar "y" ∈ FVar "s"); [ | apply R_Or_i1; apply R_Ax; compute; intuition ]. (* set (L := _ ∈ _ -> _). assert (Pr Intuiti ([pairing] ⊢ pairing)). { apply R_Ax. apply in_eq. } unfold pairing in H. apply R_All_e with (t := FVar "x") in H; cbn in H. apply R_All_e with (t := FVar "x") in H; cbn in H. apply R_Ex_e with (x := "s") (B := ∀ (#0 ∈ #1 -> #0 = FVar "x" \/ #0 = FVar "x") /\ (#0 = FVar "x" \/ #0 = FVar "x" -> #0 ∈ #1)) in H; [ cbn | calc | cbn ]. * apply R_All_e with (t := FVar "y") in H; cbn in H. admit. * apply R_And_i.*) Lemma union : IsTheorem Intuiti ZF (∀∀∃∀ (#0 ∈ #1 <-> #0 ∈ #2 \/ #0 ∈ #3)). Admitted. \ No newline at end of file
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