Theorem onin 6371

Description: The intersection of two ordinal numbers is an ordinal number. (Contributed by NM, 7-Apr-1995.)

Assertion

Ref	Expression
onin	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ 𝐵) ∈ On)

Proof of Theorem onin

Step	Hyp	Ref	Expression
1		eloni 6350	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		eloni 6350	. . 3 ⊢ (𝐵 ∈ On → Ord 𝐵)
3		ordin 6370	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵))
4	1, 2, 3	syl2an 605	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 ∩ 𝐵))
5		simpl 486	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
6		inex1g 5272	. . 3 ⊢ (𝐴 ∈ On → (𝐴 ∩ 𝐵) ∈ V)
7		elong 6348	. . 3 ⊢ ((𝐴 ∩ 𝐵) ∈ V → ((𝐴 ∩ 𝐵) ∈ On ↔ Ord (𝐴 ∩ 𝐵)))
8	5, 6, 7	3syl 18	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ∩ 𝐵) ∈ On ↔ Ord (𝐴 ∩ 𝐵)))
9	4, 8	mpbird 259	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ 𝐵) ∈ On)

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∈ wcel 2141  Vcvv 3453   ∩ cin 3901  Ord word 6339  Oncon0 6340

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243

This theorem depends on definitions:  df-bi 209  df-an 400  df-3an 1099  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rab 3414  df-v 3455  df-in 3909  df-ss 3919  df-uni 4863  df-tr 5205  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-ord 6343  df-on 6344

This theorem is referenced by:  tfrlem5  8343  noreson  27711  ontopbas  36748