Theorem omun 7835

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 4122	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
2		eleq1a 2835	. . . 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 243	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐵) ∈ ω))
5		ssequn2 4125	. . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∪ 𝐵) = 𝐴)
6		eleq1a 2835	. . . 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 243	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐵 ⊆ 𝐴 → (𝐴 ∪ 𝐵) ∈ ω))
9		nnord 7821	. . 3 ⊢ (𝐴 ∈ ω → Ord 𝐴)
10		nnord 7821	. . 3 ⊢ (𝐵 ∈ ω → Ord 𝐵)
11		ordtri2or2 6418	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
12	9, 10, 11	syl2an 602	. 2 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
13	4, 8, 12	mpjaod 866	1 ⊢ ((𝐴 ∈ ω ∧ 𝐵 ∈ ω) → (𝐴 ∪ 𝐵) ∈ ω)

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 853   = wceq 1547   ∈ wcel 2119   ∪ cun 3888   ⊆ wss 3890  Ord word 6316  ωcom 7813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-tr 5187  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-ord 6320  df-on 6321  df-om 7814

This theorem is referenced by:  precsexlem10  28233