Theorem onsucuni2 4668

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 2294	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → (𝐴 ∈ On ↔ suc 𝐵 ∈ On))
2	1	biimpac 298	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐵 ∈ On)
3		onsucb 4607	. . . . . . 7 ⊢ (𝐵 ∈ On ↔ suc 𝐵 ∈ On)
4		eloni 4478	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → Ord 𝐵)
5		ordtr 4481	. . . . . . . . . 10 ⊢ (Ord 𝐵 → Tr 𝐵)
6	4, 5	syl 14	. . . . . . . . 9 ⊢ (𝐵 ∈ On → Tr 𝐵)
7		unisucg 4517	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (Tr 𝐵 ↔ ∪ suc 𝐵 = 𝐵))
8	6, 7	mpbid 147	. . . . . . . 8 ⊢ (𝐵 ∈ On → ∪ suc 𝐵 = 𝐵)
9		suceq 4505	. . . . . . . 8 ⊢ (∪ suc 𝐵 = 𝐵 → suc ∪ suc 𝐵 = suc 𝐵)
10	8, 9	syl 14	. . . . . . 7 ⊢ (𝐵 ∈ On → suc ∪ suc 𝐵 = suc 𝐵)
11	3, 10	sylbir 135	. . . . . 6 ⊢ (suc 𝐵 ∈ On → suc ∪ suc 𝐵 = suc 𝐵)
12		eloni 4478	. . . . . . . 8 ⊢ (suc 𝐵 ∈ On → Ord suc 𝐵)
13		ordtr 4481	. . . . . . . 8 ⊢ (Ord suc 𝐵 → Tr suc 𝐵)
14	12, 13	syl 14	. . . . . . 7 ⊢ (suc 𝐵 ∈ On → Tr suc 𝐵)
15		unisucg 4517	. . . . . . 7 ⊢ (suc 𝐵 ∈ On → (Tr suc 𝐵 ↔ ∪ suc suc 𝐵 = suc 𝐵))
16	14, 15	mpbid 147	. . . . . 6 ⊢ (suc 𝐵 ∈ On → ∪ suc suc 𝐵 = suc 𝐵)
17	11, 16	eqtr4d 2267	. . . . 5 ⊢ (suc 𝐵 ∈ On → suc ∪ suc 𝐵 = ∪ suc suc 𝐵)
18	2, 17	syl 14	. . . 4 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ suc 𝐵 = ∪ suc suc 𝐵)
19		unieq 3907	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → ∪ 𝐴 = ∪ suc 𝐵)
20		suceq 4505	. . . . . 6 ⊢ (∪ 𝐴 = ∪ suc 𝐵 → suc ∪ 𝐴 = suc ∪ suc 𝐵)
21	19, 20	syl 14	. . . . 5 ⊢ (𝐴 = suc 𝐵 → suc ∪ 𝐴 = suc ∪ suc 𝐵)
22		suceq 4505	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
23	22	unieqd 3909	. . . . 5 ⊢ (𝐴 = suc 𝐵 → ∪ suc 𝐴 = ∪ suc suc 𝐵)
24	21, 23	eqeq12d 2246	. . . 4 ⊢ (𝐴 = suc 𝐵 → (suc ∪ 𝐴 = ∪ suc 𝐴 ↔ suc ∪ suc 𝐵 = ∪ suc suc 𝐵))
25	18, 24	imbitrrid 156	. . 3 ⊢ (𝐴 = suc 𝐵 → ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = ∪ suc 𝐴))
26	25	anabsi7 583	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = ∪ suc 𝐴)
27		eloni 4478	. . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴)
28		ordtr 4481	. . . . 5 ⊢ (Ord 𝐴 → Tr 𝐴)
29	27, 28	syl 14	. . . 4 ⊢ (𝐴 ∈ On → Tr 𝐴)
30		unisucg 4517	. . . 4 ⊢ (𝐴 ∈ On → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴))
31	29, 30	mpbid 147	. . 3 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)
32	31	adantr 276	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → ∪ suc 𝐴 = 𝐴)
33	26, 32	eqtrd 2264	1 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398   ∈ wcel 2202  ∪ cuni 3898  Tr wtr 4192  Ord word 4465  Oncon0 4466  suc csuc 4468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-uni 3899  df-tr 4193  df-iord 4469  df-on 4471  df-suc 4474

This theorem is referenced by:  nnsucpred  4721  nnpredcl  4727