Theorem onsucuni2 7778

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 2825	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → (𝐴 ∈ On ↔ suc 𝐵 ∈ On))
2	1	biimpac 478	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐵 ∈ On)
3		eloni 6327	. . . . 5 ⊢ (suc 𝐵 ∈ On → Ord suc 𝐵)
4		ordsuc 7758	. . . . . . . 8 ⊢ (Ord 𝐵 ↔ Ord suc 𝐵)
5		ordunisuc 7776	. . . . . . . 8 ⊢ (Ord 𝐵 → ∪ suc 𝐵 = 𝐵)
6	4, 5	sylbir 235	. . . . . . 7 ⊢ (Ord suc 𝐵 → ∪ suc 𝐵 = 𝐵)
7		suceq 6385	. . . . . . 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 7776	. . . . . 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 4862	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → ∪ 𝐴 = ∪ suc 𝐵)
13		suceq 6385	. . . . . 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 6385	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
16	15	unieqd 4864	. . . . 5 ⊢ (𝐴 = suc 𝐵 → ∪ suc 𝐴 = ∪ suc suc 𝐵)
17	14, 16	eqeq12d 2753	. . . 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 672	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = ∪ suc 𝐴)
20		eloni 6327	. . . 4 ⊢ (𝐴 ∈ On → Ord 𝐴)
21		ordunisuc 7776	. . . 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 2772	1 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∪ cuni 4851  Ord word 6316  Oncon0 6317  suc csuc 6319

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-tr 5194  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5577  df-we 5579  df-ord 6320  df-on 6321  df-suc 6323

This theorem is referenced by:  rankxplim3  9796  rankxpsuc  9797  onsucf1lem  43715