Theorem onsucuni2 7656

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 2826	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → (𝐴 ∈ On ↔ suc 𝐵 ∈ On))
2	1	biimpac 478	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐵 ∈ On)
3		eloni 6261	. . . . 5 ⊢ (suc 𝐵 ∈ On → Ord suc 𝐵)
4		ordsuc 7636	. . . . . . . 8 ⊢ (Ord 𝐵 ↔ Ord suc 𝐵)
5		ordunisuc 7654	. . . . . . . 8 ⊢ (Ord 𝐵 → ∪ suc 𝐵 = 𝐵)
6	4, 5	sylbir 234	. . . . . . 7 ⊢ (Ord suc 𝐵 → ∪ suc 𝐵 = 𝐵)
7		suceq 6316	. . . . . . 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 7654	. . . . . 6 ⊢ (Ord suc 𝐵 → ∪ suc suc 𝐵 = suc 𝐵)
10	8, 9	eqtr4d 2781	. . . . 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 4847	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → ∪ 𝐴 = ∪ suc 𝐵)
13		suceq 6316	. . . . . 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 6316	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
16	15	unieqd 4850	. . . . 5 ⊢ (𝐴 = suc 𝐵 → ∪ suc 𝐴 = ∪ suc suc 𝐵)
17	14, 16	eqeq12d 2754	. . . 4 ⊢ (𝐴 = suc 𝐵 → (suc ∪ 𝐴 = ∪ suc 𝐴 ↔ suc ∪ suc 𝐵 = ∪ suc suc 𝐵))
18	11, 17	syl5ibr 245	. . 3 ⊢ (𝐴 = suc 𝐵 → ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = ∪ suc 𝐴))
19	18	anabsi7 667	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = ∪ suc 𝐴)
20		eloni 6261	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
21		ordunisuc 7654	. . . 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 2778	1 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∪ cuni 4836  Ord word 6250  Oncon0 6251  suc csuc 6253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-tr 5188  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-ord 6254  df-on 6255  df-suc 6257

This theorem is referenced by:  rankxplim3  9570  rankxpsuc  9571