Theorem onun2 6467

Description: The union of two ordinals is an ordinal. (Contributed by Scott Fenton, 9-Aug-2024.)

Assertion

Ref	Expression
onun2	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On)

Proof of Theorem onun2

Step	Hyp	Ref	Expression
1		ssequn1 4166	. . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
2		eleq1a 2830	. . . 4 ⊢ (𝐵 ∈ On → ((𝐴 ∪ 𝐵) = 𝐵 → (𝐴 ∪ 𝐵) ∈ On))
3	2	adantl 481	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ∪ 𝐵) = 𝐵 → (𝐴 ∪ 𝐵) ∈ On))
4	1, 3	biimtrid 242	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐵) ∈ On))
5		ssequn2 4169	. . 3 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐴 ∪ 𝐵) = 𝐴)
6		eleq1a 2830	. . . 4 ⊢ (𝐴 ∈ On → ((𝐴 ∪ 𝐵) = 𝐴 → (𝐴 ∪ 𝐵) ∈ On))
7	6	adantr 480	. . 3 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → ((𝐴 ∪ 𝐵) = 𝐴 → (𝐴 ∪ 𝐵) ∈ On))
8	5, 7	biimtrid 242	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐵 ⊆ 𝐴 → (𝐴 ∪ 𝐵) ∈ On))
9		eloni 6367	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
10		eloni 6367	. . 3 ⊢ (𝐵 ∈ On → Ord 𝐵)
11		ordtri2or2 6458	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
12	9, 10, 11	syl2an 596	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
13	4, 8, 12	mpjaod 860	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ∪ 𝐵) ∈ On)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ∪ cun 3929   ⊆ wss 3931  Ord word 6356  Oncon0 6357

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-pss 3951  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-tr 5235  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-ord 6360  df-on 6361

This theorem is referenced by:  onun2i  6481  nosupinfsep  27701  onexlimgt  43234  omabs2  43323  onsucunitp  43364