Metamath Proof Explorer

Theorem ordtype2

Description: For any set-like well-ordered class, if the order isomorphism exists (is a set), then it maps some ordinal onto A isomorphically. Otherwise, F is a proper class, which implies that either ran F C_ A is a proper class or dom F = On . This weak version of ordtype does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 25-Jun-2015)

Ref Expression
Hypothesis oicl.1 ⊢ F = OrdIso R A
Assertion ordtype2 ⊢ R We A ∧ R Se A ∧ F ∈ V → F Isom E , R dom ⁡ F A

Proof

Step Hyp Ref Expression
1 oicl.1 ⊢ F = OrdIso R A
2 eqid ⊢ recs ⁡ h ∈ V ⟼ ι v ∈ w ∈ A | ∀ j ∈ ran ⁡ h j R w | ∀ u ∈ w ∈ A | ∀ j ∈ ran ⁡ h j R w ¬ u R v = recs ⁡ h ∈ V ⟼ ι v ∈ w ∈ A | ∀ j ∈ ran ⁡ h j R w | ∀ u ∈ w ∈ A | ∀ j ∈ ran ⁡ h j R w ¬ u R v
3 eqid ⊢ w ∈ A | ∀ j ∈ ran ⁡ h j R w = w ∈ A | ∀ j ∈ ran ⁡ h j R w
4 eqid ⊢ h ∈ V ⟼ ι v ∈ w ∈ A | ∀ j ∈ ran ⁡ h j R w | ∀ u ∈ w ∈ A | ∀ j ∈ ran ⁡ h j R w ¬ u R v = h ∈ V ⟼ ι v ∈ w ∈ A | ∀ j ∈ ran ⁡ h j R w | ∀ u ∈ w ∈ A | ∀ j ∈ ran ⁡ h j R w ¬ u R v
5 2 3 4 ordtypecbv ⊢ recs ⁡ f ∈ V ⟼ ι s ∈ y ∈ A | ∀ i ∈ ran ⁡ f i R y | ∀ r ∈ y ∈ A | ∀ i ∈ ran ⁡ f i R y ¬ r R s = recs ⁡ h ∈ V ⟼ ι v ∈ w ∈ A | ∀ j ∈ ran ⁡ h j R w | ∀ u ∈ w ∈ A | ∀ j ∈ ran ⁡ h j R w ¬ u R v
6 eqid ⊢ x ∈ On | ∃ t ∈ A ∀ z ∈ recs ⁡ f ∈ V ⟼ ι s ∈ y ∈ A | ∀ i ∈ ran ⁡ f i R y | ∀ r ∈ y ∈ A | ∀ i ∈ ran ⁡ f i R y ¬ r R s x z R t = x ∈ On | ∃ t ∈ A ∀ z ∈ recs ⁡ f ∈ V ⟼ ι s ∈ y ∈ A | ∀ i ∈ ran ⁡ f i R y | ∀ r ∈ y ∈ A | ∀ i ∈ ran ⁡ f i R y ¬ r R s x z R t
7 simp1 ⊢ R We A ∧ R Se A ∧ F ∈ V → R We A
8 simp2 ⊢ R We A ∧ R Se A ∧ F ∈ V → R Se A
9 simp3 ⊢ R We A ∧ R Se A ∧ F ∈ V → F ∈ V
10 5 3 4 6 1 7 8 9 ordtypelem9 ⊢ R We A ∧ R Se A ∧ F ∈ V → F Isom E , R dom ⁡ F A