Theorem onsucuni2 7809

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 2849	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → (𝐴 ∈ On ↔ suc 𝐵 ∈ On))
2	1	biimpac 482	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐵 ∈ On)
3		eloni 6351	. . . . 5 ⊢ (suc 𝐵 ∈ On → Ord suc 𝐵)
4		ordsuc 7789	. . . . . . . 8 ⊢ (Ord 𝐵 ↔ Ord suc 𝐵)
5		ordunisuc 7807	. . . . . . . 8 ⊢ (Ord 𝐵 → ∪ suc 𝐵 = 𝐵)
6	4, 5	sylbir 237	. . . . . . 7 ⊢ (Ord suc 𝐵 → ∪ suc 𝐵 = 𝐵)
7		suceq 6409	. . . . . . 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 7807	. . . . . 6 ⊢ (Ord suc 𝐵 → ∪ suc suc 𝐵 = suc 𝐵)
10	8, 9	eqtr4d 2799	. . . . 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 4873	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → ∪ 𝐴 = ∪ suc 𝐵)
13		suceq 6409	. . . . . 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 6409	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
16	15	unieqd 4875	. . . . 5 ⊢ (𝐴 = suc 𝐵 → ∪ suc 𝐴 = ∪ suc suc 𝐵)
17	14, 16	eqeq12d 2777	. . . 4 ⊢ (𝐴 = suc 𝐵 → (suc ∪ 𝐴 = ∪ suc 𝐴 ↔ suc ∪ suc 𝐵 = ∪ suc suc 𝐵))
18	11, 17	imbitrrid 248	. . 3 ⊢ (𝐴 = suc 𝐵 → ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = ∪ suc 𝐴))
19	18	anabsi7 681	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = ∪ suc 𝐴)
20		eloni 6351	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
21		ordunisuc 7807	. . . 4 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴)
22	20, 21	syl 17	. . 3 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)
23	22	adantr 484	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → ∪ suc 𝐴 = 𝐴)
24	19, 23	eqtrd 2796	1 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∪ cuni 4862  Ord word 6340  Oncon0 6341  suc csuc 6343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243  ax-pr 5387  ax-un 7713

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-tr 5205  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-ord 6344  df-on 6345  df-suc 6347

This theorem is referenced by:  rankxplim3  9833  rankxpsuc  9834  onsucf1lem  43807