Theorem onun2i 5760

Description: The union of two ordinal numbers is an ordinal number. (Contributed by NM, 13-Jun-1994.)

Hypotheses

Ref	Expression
on.1	⊢ 𝐴 ∈ On
on.2	⊢ 𝐵 ∈ On

Assertion

Ref	Expression
onun2i	⊢ (𝐴 ∪ 𝐵) ∈ On

Proof of Theorem onun2i

Step	Hyp	Ref	Expression
1		on.2	. . . 4 ⊢ 𝐵 ∈ On
2	1	onordi 5749	. . 3 ⊢ Ord 𝐵
3		on.1	. . . 4 ⊢ 𝐴 ∈ On
4	3	onordi 5749	. . 3 ⊢ Ord 𝐴
5		ordtri2or 5739	. . 3 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵 ∈ 𝐴 ∨ 𝐴 ⊆ 𝐵))
6	2, 4, 5	mp2an 704	. 2 ⊢ (𝐵 ∈ 𝐴 ∨ 𝐴 ⊆ 𝐵)
7	3	oneluni 5757	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (𝐴 ∪ 𝐵) = 𝐴)
8	7, 3	syl6eqel 2696	. . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐴 ∪ 𝐵) ∈ On)
9		ssequn1 3745	. . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
10		eleq1 2676	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) = 𝐵 → ((𝐴 ∪ 𝐵) ∈ On ↔ 𝐵 ∈ On))
11	1, 10	mpbiri 247	. . . 4 ⊢ ((𝐴 ∪ 𝐵) = 𝐵 → (𝐴 ∪ 𝐵) ∈ On)
12	9, 11	sylbi 206	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐵) ∈ On)
13	8, 12	jaoi 393	. 2 ⊢ ((𝐵 ∈ 𝐴 ∨ 𝐴 ⊆ 𝐵) → (𝐴 ∪ 𝐵) ∈ On)
14	6, 13	ax-mp 5	1 ⊢ (𝐴 ∪ 𝐵) ∈ On

Syntax hints:   ∨ wo 382   = wceq 1475   ∈ wcel 1977   ∪ cun 3538   ⊆ wss 3540  Ord word 5639  Oncon0 5640

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-tr 4681  df-eprel 4949  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-ord 5643  df-on 5644

This theorem is referenced by:  rankunb  8596  rankelun  8618  rankelpr  8619  inar1  9476