Theorem omun 7899

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 4184	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
2		eleq1a 2855	. . . 4 ⊢ (𝐵 ∈ ω → ((𝐴 ∪ 𝐵) = 𝐵 → (𝐴 ∪ 𝐵) ∈ ω))
3	2	adantl 487	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ∪ 𝐵) = 𝐵 → (𝐴 ∪ 𝐵) ∈ ω))
4	1, 3	biimtrid 245	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐵) ∈ ω))
5		ssequn2 4187	. . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∪ 𝐵) = 𝐴)
6		eleq1a 2855	. . . 4 ⊢ (𝐴 ∈ ω → ((𝐴 ∪ 𝐵) = 𝐴 → (𝐴 ∪ 𝐵) ∈ ω))
7	6	adantr 486	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ∪ 𝐵) = 𝐴 → (𝐴 ∪ 𝐵) ∈ ω))
8	5, 7	biimtrid 245	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ⊆ 𝐴 → (𝐴 ∪ 𝐵) ∈ ω))
9		nnord 7885	. . 3 ⊢ (𝐴 ∈ ω → Ord 𝐴)
10		nnord 7885	. . 3 ⊢ (𝐵 ∈ ω → Ord 𝐵)
11		ordtri2or2 6464	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
12	9, 10, 11	syl2an 608	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
13	4, 8, 12	mpjaod 874	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∪ 𝐵) ∈ ω)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895   ∪ cun 4149  Ord word 6361  ωcom 7877

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-ord 6365  df-on 6366  df-om 7878

This theorem is used by:  precsexlem10  28607