Theorem onin 6383

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 6362	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		eloni 6362	. . 3 ⊢ (𝐵 ∈ On → Ord 𝐵)
3		ordin 6382	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵))
4	1, 2, 3	syl2an 596	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → Ord (𝐴 ∩ 𝐵))
5		simpl 482	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → 𝐴 ∈ On)
6		inex1g 5289	. . 3 ⊢ (𝐴 ∈ On → (𝐴 ∩ 𝐵) ∈ V)
7		elong 6360	. . 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 257	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∩ 𝐵) ∈ On)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  Vcvv 3459   ∩ cin 3925  Ord word 6351  Oncon0 6352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rab 3416  df-v 3461  df-in 3933  df-ss 3943  df-uni 4884  df-tr 5230  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-ord 6355  df-on 6356

This theorem is referenced by:  tfrlem5  8394  noreson  27624  ontopbas  36446