Theorem omun 7874

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 4179	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
2		eleq1a 2828	. . . 4 ⊢ (𝐵 ∈ ω → ((𝐴 ∪ 𝐵) = 𝐵 → (𝐴 ∪ 𝐵) ∈ ω))
3	2	adantl 482	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ∪ 𝐵) = 𝐵 → (𝐴 ∪ 𝐵) ∈ ω))
4	1, 3	biimtrid 241	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐵) ∈ ω))
5		ssequn2 4182	. . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∪ 𝐵) = 𝐴)
6		eleq1a 2828	. . . 4 ⊢ (𝐴 ∈ ω → ((𝐴 ∪ 𝐵) = 𝐴 → (𝐴 ∪ 𝐵) ∈ ω))
7	6	adantr 481	. . 3 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → ((𝐴 ∪ 𝐵) = 𝐴 → (𝐴 ∪ 𝐵) ∈ ω))
8	5, 7	biimtrid 241	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ⊆ 𝐴 → (𝐴 ∪ 𝐵) ∈ ω))
9		nnord 7859	. . 3 ⊢ (𝐴 ∈ ω → Ord 𝐴)
10		nnord 7859	. . 3 ⊢ (𝐵 ∈ ω → Ord 𝐵)
11		ordtri2or2 6460	. . 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 858	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∪ 𝐵) ∈ ω)

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106   ∪ cun 3945   ⊆ wss 3947  Ord word 6360  ωcom 7851

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6364  df-on 6365  df-om 7852

This theorem is referenced by:  precsexlem10  27651