Theorem onun2i 6305

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 6294	. . 3 ⊢ Ord 𝐵
3		on.1	. . . 4 ⊢ 𝐴 ∈ On
4	3	onordi 6294	. . 3 ⊢ Ord 𝐴
5		ordtri2or 6285	. . 3 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵 ∈ 𝐴 ∨ 𝐴 ⊆ 𝐵))
6	2, 4, 5	mp2an 690	. 2 ⊢ (𝐵 ∈ 𝐴 ∨ 𝐴 ⊆ 𝐵)
7	3	oneluni 6302	. . . 4 ⊢ (𝐵 ∈ 𝐴 → (𝐴 ∪ 𝐵) = 𝐴)
8	7, 3	eqeltrdi 2921	. . 3 ⊢ (𝐵 ∈ 𝐴 → (𝐴 ∪ 𝐵) ∈ On)
9		ssequn1 4155	. . . 4 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∪ 𝐵) = 𝐵)
10		eleq1 2900	. . . . 5 ⊢ ((𝐴 ∪ 𝐵) = 𝐵 → ((𝐴 ∪ 𝐵) ∈ On ↔ 𝐵 ∈ On))
11	1, 10	mpbiri 260	. . . 4 ⊢ ((𝐴 ∪ 𝐵) = 𝐵 → (𝐴 ∪ 𝐵) ∈ On)
12	9, 11	sylbi 219	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∪ 𝐵) ∈ On)
13	8, 12	jaoi 853	. 2 ⊢ ((𝐵 ∈ 𝐴 ∨ 𝐴 ⊆ 𝐵) → (𝐴 ∪ 𝐵) ∈ On)
14	6, 13	ax-mp 5	1 ⊢ (𝐴 ∪ 𝐵) ∈ On

Syntax hints:   ∨ wo 843   = wceq 1533   ∈ wcel 2110   ∪ cun 3933   ⊆ wss 3935  Ord word 6189  Oncon0 6190

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-tr 5172  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-ord 6193  df-on 6194

This theorem is referenced by:  rankunb  9278  rankelun  9300  rankelpr  9301  inar1  10196