Commit debb9b68 by Samuel Ben Hamou

### Fin preuve union.

parent 50553df9
 ... ... @@ -236,12 +236,14 @@ Qed. Lemma union : IsTheorem Intuiti ZF (∀∀∃∀ (#0 ∈ #1 <-> #0 ∈ #3 \/ #0 ∈ #2)). Proof. thm. exists [ pairing; union ]. exists [ pairing; union; compat_right; eq_refl ]. split; auto. - simpl. constructor. + left. calc. + constructor; [ left; calc | auto ]. - set (Γ := [ _ ; _ ]). + constructor. left. calc. constructor. left. calc. constructor; [ left; calc | auto ]. - set (Γ := [ _ ; _ ; _ ]). app_R_All_i "A" A. app_R_All_i "B" B. inst_axiom pairing [A; B]; cbn in *. fold A. fold B. fold A in H. fold B in H. reIff. ... ... @@ -272,11 +274,39 @@ Proof. set (Ax := ∀ _ <-> _ \/ _). inst_axiom Ax [ y ]. exact H. -- apply R_Or_i1. (* todo avec compat_left *) admit. apply R_Imp_e with (A := x ∈ y /\ y = A). ++ inst_axiom compat_right [ A; y; x ]. ++ apply R_And_i; apply R_Ax; calc. -- apply R_Or_i2. (* todo *) admit. apply R_Imp_e with (A := x ∈ y /\ y = B). ++ inst_axiom compat_right [ B; y; x ]. ++ apply R_And_i; apply R_Ax; calc. + apply R_Imp_i. apply R'_Or_e. * admit. \ No newline at end of file * set (Ax := ∀ _ ∈ U <-> _). inst_axiom Ax [ x ]. cbn in H. fold U in H. fold C in H. fold x in H. fold Ax in H. apply R_And_e2 in H. apply R_Imp_e with (A := (∃ #0 ∈ C /\ x ∈ #0)); [ assumption | ]. apply R_Ex_i with (t := A). cbn. fold C. fold A. fold x. apply R_And_i. -- set (Ax2 := ∀ _ <-> _). inst_axiom Ax2 [ A ]. cbn in H0. fold B in H0. fold C in H0. fold A in H0. fold Ax in H0. apply R_And_e2 in H0. apply R_Imp_e with (A := A = A \/ A = B); [ assumption | ]. apply R_Or_i1. inst_axiom eq_refl [ A ]. -- apply R'_Ax. * set (Ax := ∀ _ ∈ U <-> _). inst_axiom Ax [ x ]. cbn in H. fold U in H. fold C in H. fold x in H. fold Ax in H. apply R_And_e2 in H. apply R_Imp_e with (A := (∃ #0 ∈ C /\ x ∈ #0)); [ assumption | ]. apply R_Ex_i with (t := B). cbn. fold C. fold B. fold x. apply R_And_i. -- set (Ax2 := ∀ _ <-> _). inst_axiom Ax2 [ B ]. cbn in H0. fold B in H0. fold C in H0. fold A in H0. fold Ax in H0. apply R_And_e2 in H0. apply R_Imp_e with (A := B = A \/ B = B); [ assumption | ]. apply R_Or_i2. inst_axiom eq_refl [ B ]. -- apply R'_Ax. Qed. \ No newline at end of file
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