Theorem onsucuni2 4600

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 2259	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → (𝐴 ∈ On ↔ suc 𝐵 ∈ On))
2	1	biimpac 298	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐵 ∈ On)
3		onsucb 4539	. . . . . . 7 ⊢ (𝐵 ∈ On ↔ suc 𝐵 ∈ On)
4		eloni 4410	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → Ord 𝐵)
5		ordtr 4413	. . . . . . . . . 10 ⊢ (Ord 𝐵 → Tr 𝐵)
6	4, 5	syl 14	. . . . . . . . 9 ⊢ (𝐵 ∈ On → Tr 𝐵)
7		unisucg 4449	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (Tr 𝐵 ↔ ∪ suc 𝐵 = 𝐵))
8	6, 7	mpbid 147	. . . . . . . 8 ⊢ (𝐵 ∈ On → ∪ suc 𝐵 = 𝐵)
9		suceq 4437	. . . . . . . 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 4410	. . . . . . . 8 ⊢ (suc 𝐵 ∈ On → Ord suc 𝐵)
13		ordtr 4413	. . . . . . . 8 ⊢ (Ord suc 𝐵 → Tr suc 𝐵)
14	12, 13	syl 14	. . . . . . 7 ⊢ (suc 𝐵 ∈ On → Tr suc 𝐵)
15		unisucg 4449	. . . . . . 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 2232	. . . . 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 3848	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → ∪ 𝐴 = ∪ suc 𝐵)
20		suceq 4437	. . . . . 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 4437	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
23	22	unieqd 3850	. . . . 5 ⊢ (𝐴 = suc 𝐵 → ∪ suc 𝐴 = ∪ suc suc 𝐵)
24	21, 23	eqeq12d 2211	. . . 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 581	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = ∪ suc 𝐴)
27		eloni 4410	. . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴)
28		ordtr 4413	. . . . 5 ⊢ (Ord 𝐴 → Tr 𝐴)
29	27, 28	syl 14	. . . 4 ⊢ (𝐴 ∈ On → Tr 𝐴)
30		unisucg 4449	. . . 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 2229	1 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364   ∈ wcel 2167  ∪ cuni 3839  Tr wtr 4131  Ord word 4397  Oncon0 4398  suc csuc 4400

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4151  ax-pow 4207  ax-pr 4242  ax-un 4468

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3607  df-sn 3628  df-pr 3629  df-uni 3840  df-tr 4132  df-iord 4401  df-on 4403  df-suc 4406

This theorem is referenced by:  nnsucpred  4653  nnpredcl  4659