Theorem onun2i 5843

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 5832	. . 3 ⊢ Ord 𝐵
3		on.1	. . . 4 ⊢ 𝐴 ∈ On
4	3	onordi 5832	. . 3 ⊢ Ord 𝐴
5		ordtri2or 5822	. . 3 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵 ∈ 𝐴 ∨ 𝐴 ⊆ 𝐵))
6	2, 4, 5	mp2an 708	. 2 ⊢ (𝐵 ∈ 𝐴 ∨ 𝐴 ⊆ 𝐵)
7	3	oneluni 5840	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (𝐴 ∪ 𝐵) = 𝐴)
8	7, 3	syl6eqel 2709	. . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐴 ∪ 𝐵) ∈ On)
9		ssequn1 3783	. . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
10		eleq1 2689	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) = 𝐵 → ((𝐴 ∪ 𝐵) ∈ On ↔ 𝐵 ∈ On))
11	1, 10	mpbiri 248	. . . 4 ⊢ ((𝐴 ∪ 𝐵) = 𝐵 → (𝐴 ∪ 𝐵) ∈ On)
12	9, 11	sylbi 207	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐵) ∈ On)
13	8, 12	jaoi 394	. 2 ⊢ ((𝐵 ∈ 𝐴 ∨ 𝐴 ⊆ 𝐵) → (𝐴 ∪ 𝐵) ∈ On)
14	6, 13	ax-mp 5	1 ⊢ (𝐴 ∪ 𝐵) ∈ On

Syntax hints:   ∨ wo 383   = wceq 1483   ∈ wcel 1990   ∪ cun 3572   ⊆ wss 3574  Ord word 5722  Oncon0 5723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727

This theorem is referenced by:  rankunb  8713  rankelun  8735  rankelpr  8736  inar1  9597