Theorem onin 6282

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 6261	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		eloni 6261	. . 3 ⊢ (𝐵 ∈ On → Ord 𝐵)
3		ordin 6281	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵))
4	1, 2, 3	syl2an 595	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 ∩ 𝐵))
5		simpl 482	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
6		inex1g 5238	. . 3 ⊢ (𝐴 ∈ On → (𝐴 ∩ 𝐵) ∈ V)
7		elong 6259	. . 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 256	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ 𝐵) ∈ On)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2108  Vcvv 3422   ∩ cin 3882  Ord word 6250  Oncon0 6251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-ext 2709  ax-sep 5218

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rab 3072  df-v 3424  df-in 3890  df-ss 3900  df-uni 4837  df-tr 5188  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-ord 6254  df-on 6255

This theorem is referenced by:  tfrlem5  8182  noreson  33790  ontopbas  34544