Theorem onin 6426

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 6405	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
2		eloni 6405	. . 3 ⊢ (𝐵 ∈ On → Ord 𝐵)
3		ordin 6425	. . 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 5337	. . 3 ⊢ (𝐴 ∈ On → (𝐴 ∩ 𝐵) ∈ V)
7		elong 6403	. . 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 3488   ∩ cin 3975  Ord word 6394  Oncon0 6395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1089  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rab 3444  df-v 3490  df-in 3983  df-ss 3993  df-uni 4932  df-tr 5284  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-ord 6398  df-on 6399

This theorem is referenced by:  tfrlem5  8436  noreson  27723  ontopbas  36394