Theorem onsucuni2 7818

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 2821	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → (𝐴 ∈ On ↔ suc 𝐵 ∈ On))
2	1	biimpac 479	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐵 ∈ On)
3		eloni 6371	. . . . 5 ⊢ (suc 𝐵 ∈ On → Ord suc 𝐵)
4		ordsuc 7797	. . . . . . . 8 ⊢ (Ord 𝐵 ↔ Ord suc 𝐵)
5		ordunisuc 7816	. . . . . . . 8 ⊢ (Ord 𝐵 → ∪ suc 𝐵 = 𝐵)
6	4, 5	sylbir 234	. . . . . . 7 ⊢ (Ord suc 𝐵 → ∪ suc 𝐵 = 𝐵)
7		suceq 6427	. . . . . . 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 7816	. . . . . 6 ⊢ (Ord suc 𝐵 → ∪ suc suc 𝐵 = suc 𝐵)
10	8, 9	eqtr4d 2775	. . . . 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 4918	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → ∪ 𝐴 = ∪ suc 𝐵)
13		suceq 6427	. . . . . 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 6427	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
16	15	unieqd 4921	. . . . 5 ⊢ (𝐴 = suc 𝐵 → ∪ suc 𝐴 = ∪ suc suc 𝐵)
17	14, 16	eqeq12d 2748	. . . 4 ⊢ (𝐴 = suc 𝐵 → (suc ∪ 𝐴 = ∪ suc 𝐴 ↔ suc ∪ suc 𝐵 = ∪ suc suc 𝐵))
18	11, 17	imbitrrid 245	. . 3 ⊢ (𝐴 = suc 𝐵 → ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = ∪ suc 𝐴))
19	18	anabsi7 669	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = ∪ suc 𝐴)
20		eloni 6371	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
21		ordunisuc 7816	. . . 4 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴)
22	20, 21	syl 17	. . 3 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)
23	22	adantr 481	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → ∪ suc 𝐴 = 𝐴)
24	19, 23	eqtrd 2772	1 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∪ cuni 4907  Ord word 6360  Oncon0 6361  suc csuc 6363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6364  df-on 6365  df-suc 6367

This theorem is referenced by:  rankxplim3  9872  rankxpsuc  9873  onsucf1lem  42004