Theorem onsucuni2 4541

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 2229	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → (𝐴 ∈ On ↔ suc 𝐵 ∈ On))
2	1	biimpac 296	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐵 ∈ On)
3		sucelon 4480	. . . . . . 7 ⊢ (𝐵 ∈ On ↔ suc 𝐵 ∈ On)
4		eloni 4353	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → Ord 𝐵)
5		ordtr 4356	. . . . . . . . . 10 ⊢ (Ord 𝐵 → Tr 𝐵)
6	4, 5	syl 14	. . . . . . . . 9 ⊢ (𝐵 ∈ On → Tr 𝐵)
7		unisucg 4392	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (Tr 𝐵 ↔ ∪ suc 𝐵 = 𝐵))
8	6, 7	mpbid 146	. . . . . . . 8 ⊢ (𝐵 ∈ On → ∪ suc 𝐵 = 𝐵)
9		suceq 4380	. . . . . . . 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 134	. . . . . 6 ⊢ (suc 𝐵 ∈ On → suc ∪ suc 𝐵 = suc 𝐵)
12		eloni 4353	. . . . . . . 8 ⊢ (suc 𝐵 ∈ On → Ord suc 𝐵)
13		ordtr 4356	. . . . . . . 8 ⊢ (Ord suc 𝐵 → Tr suc 𝐵)
14	12, 13	syl 14	. . . . . . 7 ⊢ (suc 𝐵 ∈ On → Tr suc 𝐵)
15		unisucg 4392	. . . . . . 7 ⊢ (suc 𝐵 ∈ On → (Tr suc 𝐵 ↔ ∪ suc suc 𝐵 = suc 𝐵))
16	14, 15	mpbid 146	. . . . . 6 ⊢ (suc 𝐵 ∈ On → ∪ suc suc 𝐵 = suc 𝐵)
17	11, 16	eqtr4d 2201	. . . . 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 3798	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → ∪ 𝐴 = ∪ suc 𝐵)
20		suceq 4380	. . . . . 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 4380	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
23	22	unieqd 3800	. . . . 5 ⊢ (𝐴 = suc 𝐵 → ∪ suc 𝐴 = ∪ suc suc 𝐵)
24	21, 23	eqeq12d 2180	. . . 4 ⊢ (𝐴 = suc 𝐵 → (suc ∪ 𝐴 = ∪ suc 𝐴 ↔ suc ∪ suc 𝐵 = ∪ suc suc 𝐵))
25	18, 24	syl5ibr 155	. . 3 ⊢ (𝐴 = suc 𝐵 → ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = ∪ suc 𝐴))
26	25	anabsi7 571	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = ∪ suc 𝐴)
27		eloni 4353	. . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴)
28		ordtr 4356	. . . . 5 ⊢ (Ord 𝐴 → Tr 𝐴)
29	27, 28	syl 14	. . . 4 ⊢ (𝐴 ∈ On → Tr 𝐴)
30		unisucg 4392	. . . 4 ⊢ (𝐴 ∈ On → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴))
31	29, 30	mpbid 146	. . 3 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)
32	31	adantr 274	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → ∪ suc 𝐴 = 𝐴)
33	26, 32	eqtrd 2198	1 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1343   ∈ wcel 2136  ∪ cuni 3789  Tr wtr 4080  Ord word 4340  Oncon0 4341  suc csuc 4343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-tr 4081  df-iord 4344  df-on 4346  df-suc 4349

This theorem is referenced by:  nnsucpred  4594  nnpredcl  4600