Theorem onun 2358

Description: The union of two ordinal numbers is an ordinal number.

Hypotheses

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

Assertion

Ref	Expression
onun	⊢ (A ∪ B) ∈ On

Proof of Theorem onun

Step	Hyp	Ref	Expression
1		on.2	. . . 4 ⊢ B ∈ On
2	1	onord 2343	. . 3 ⊢ Ord B
3		on.1	. . . 4 ⊢ A ∈ On
4	3	onord 2343	. . 3 ⊢ Ord A
5		ordtri2or 2328	. . 3 ⊢ ((Ord B ∧ Ord A) → (B ∈ A ∨ A ⊆ B))
6	2, 4, 5	mp2an 520	. 2 ⊢ (B ∈ A ∨ A ⊆ B)
7	3	onelun 2352	. . . 4 ⊢ (B ∈ A → (A ∪ B) = A)
8		eleq1 1149	. . . . 5 ⊢ ((A ∪ B) = A → ((A ∪ B) ∈ On ↔ A ∈ On))
9	3, 8	mpbiri 169	. . . 4 ⊢ ((A ∪ B) = A → (A ∪ B) ∈ On)
10	7, 9	syl 12	. . 3 ⊢ (B ∈ A → (A ∪ B) ∈ On)
11		ssequn1 1628	. . . 4 ⊢ (A ⊆ B ↔ (A ∪ B) = B)
12		eleq1 1149	. . . . 5 ⊢ ((A ∪ B) = B → ((A ∪ B) ∈ On ↔ B ∈ On))
13	1, 12	mpbiri 169	. . . 4 ⊢ ((A ∪ B) = B → (A ∪ B) ∈ On)
14	11, 13	sylbi 174	. . 3 ⊢ (A ⊆ B → (A ∪ B) ∈ On)
15	10, 14	jaoi 275	. 2 ⊢ ((B ∈ A ∨ A ⊆ B) → (A ∪ B) ∈ On)
16	6, 15	ax-mp 6	1 ⊢ (A ∪ B) ∈ On

Syntax hints:   ∨ wo 195   = wceq 1091   ∈ wcel 1092   ∪ cun 1485   ⊆ wss 1487  Ord word 2198  Oncon0 2199

This theorem is referenced by:  rankun 3535

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203

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