Theorem onin 6060

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 6039	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		eloni 6039	. . 3 ⊢ (𝐵 ∈ On → Ord 𝐵)
3		ordin 6059	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵))
4	1, 2, 3	syl2an 586	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 ∩ 𝐵))
5		simpl 475	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
6		inex1g 5080	. . 3 ⊢ (𝐴 ∈ On → (𝐴 ∩ 𝐵) ∈ V)
7		elong 6037	. . 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 249	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ 𝐵) ∈ On)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   ∈ wcel 2050  Vcvv 3416   ∩ cin 3829  Ord word 6028  Oncon0 6029

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-ext 2751  ax-sep 5060

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2760  df-cleq 2772  df-clel 2847  df-nfc 2919  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3418  df-in 3837  df-ss 3844  df-uni 4713  df-tr 5031  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-ord 6032  df-on 6033

This theorem is referenced by:  tfrlem5  7820  noreson  32685  ontopbas  33293