Theorem omun 7910

Description: The union of two finite ordinals is a finite ordinal. (Contributed by Scott Fenton, 15-Mar-2025.)

Assertion

Ref	Expression
omun	⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∪ 𝐵) ∈ ω)

Proof of Theorem omun

Step	Hyp	Ref	Expression
1		ssequn1 4185	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
2		eleq1a 2835	. . . 4 ⊢ (𝐵 ∈ ω → ((𝐴 ∪ 𝐵) = 𝐵 → (𝐴 ∪ 𝐵) ∈ ω))
3	2	adantl 481	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ∪ 𝐵) = 𝐵 → (𝐴 ∪ 𝐵) ∈ ω))
4	1, 3	biimtrid 242	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐵) ∈ ω))
5		ssequn2 4188	. . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∪ 𝐵) = 𝐴)
6		eleq1a 2835	. . . 4 ⊢ (𝐴 ∈ ω → ((𝐴 ∪ 𝐵) = 𝐴 → (𝐴 ∪ 𝐵) ∈ ω))
7	6	adantr 480	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ∪ 𝐵) = 𝐴 → (𝐴 ∪ 𝐵) ∈ ω))
8	5, 7	biimtrid 242	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ⊆ 𝐴 → (𝐴 ∪ 𝐵) ∈ ω))
9		nnord 7896	. . 3 ⊢ (𝐴 ∈ ω → Ord 𝐴)
10		nnord 7896	. . 3 ⊢ (𝐵 ∈ ω → Ord 𝐵)
11		ordtri2or2 6482	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
12	9, 10, 11	syl2an 596	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
13	4, 8, 12	mpjaod 860	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∪ 𝐵) ∈ ω)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1539   ∈ wcel 2107   ∪ cun 3948   ⊆ wss 3950  Ord word 6382  ωcom 7888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-tr 5259  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-ord 6386  df-on 6387  df-om 7889

This theorem is referenced by:  precsexlem10  28241  zs12bday  28425