Theorem onsucuni2 7833

Description: A successor ordinal is the successor of its union. (Contributed by NM, 10-Dec-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
onsucuni2	⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = 𝐴)

Proof of Theorem onsucuni2

Step	Hyp	Ref	Expression
1		eleq1 2823	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → (𝐴 ∈ On ↔ suc 𝐵 ∈ On))
2	1	biimpac 478	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐵 ∈ On)
3		eloni 6367	. . . . 5 ⊢ (suc 𝐵 ∈ On → Ord suc 𝐵)
4		ordsuc 7812	. . . . . . . 8 ⊢ (Ord 𝐵 ↔ Ord suc 𝐵)
5		ordunisuc 7831	. . . . . . . 8 ⊢ (Ord 𝐵 → ∪ suc 𝐵 = 𝐵)
6	4, 5	sylbir 235	. . . . . . 7 ⊢ (Ord suc 𝐵 → ∪ suc 𝐵 = 𝐵)
7		suceq 6424	. . . . . . 7 ⊢ (∪ suc 𝐵 = 𝐵 → suc ∪ suc 𝐵 = suc 𝐵)
8	6, 7	syl 17	. . . . . 6 ⊢ (Ord suc 𝐵 → suc ∪ suc 𝐵 = suc 𝐵)
9		ordunisuc 7831	. . . . . 6 ⊢ (Ord suc 𝐵 → ∪ suc suc 𝐵 = suc 𝐵)
10	8, 9	eqtr4d 2774	. . . . 5 ⊢ (Ord suc 𝐵 → suc ∪ suc 𝐵 = ∪ suc suc 𝐵)
11	2, 3, 10	3syl 18	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ suc 𝐵 = ∪ suc suc 𝐵)
12		unieq 4899	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → ∪ 𝐴 = ∪ suc 𝐵)
13		suceq 6424	. . . . . 6 ⊢ (∪ 𝐴 = ∪ suc 𝐵 → suc ∪ 𝐴 = suc ∪ suc 𝐵)
14	12, 13	syl 17	. . . . 5 ⊢ (𝐴 = suc 𝐵 → suc ∪ 𝐴 = suc ∪ suc 𝐵)
15		suceq 6424	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
16	15	unieqd 4901	. . . . 5 ⊢ (𝐴 = suc 𝐵 → ∪ suc 𝐴 = ∪ suc suc 𝐵)
17	14, 16	eqeq12d 2752	. . . 4 ⊢ (𝐴 = suc 𝐵 → (suc ∪ 𝐴 = ∪ suc 𝐴 ↔ suc ∪ suc 𝐵 = ∪ suc suc 𝐵))
18	11, 17	imbitrrid 246	. . 3 ⊢ (𝐴 = suc 𝐵 → ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = ∪ suc 𝐴))
19	18	anabsi7 671	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = ∪ suc 𝐴)
20		eloni 6367	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
21		ordunisuc 7831	. . . 4 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴)
22	20, 21	syl 17	. . 3 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)
23	22	adantr 480	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → ∪ suc 𝐴 = 𝐴)
24	19, 23	eqtrd 2771	1 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∪ cuni 4888  Ord word 6356  Oncon0 6357  suc csuc 6359

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  ax-un 7734

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  df-suc 6363

This theorem is referenced by:  rankxplim3  9900  rankxpsuc  9901  onsucf1lem  43260