Theorem onsucuni2 4307

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 2141	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → (𝐴 ∈ On ↔ suc 𝐵 ∈ On))
2	1	biimpac 292	. . . . 5 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐵 ∈ On)
3		sucelon 4247	. . . . . . 7 ⊢ (𝐵 ∈ On ↔ suc 𝐵 ∈ On)
4		eloni 4130	. . . . . . . . . 10 ⊢ (𝐵 ∈ On → Ord 𝐵)
5		ordtr 4133	. . . . . . . . . 10 ⊢ (Ord 𝐵 → Tr 𝐵)
6	4, 5	syl 14	. . . . . . . . 9 ⊢ (𝐵 ∈ On → Tr 𝐵)
7		unisucg 4169	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (Tr 𝐵 ↔ ∪ suc 𝐵 = 𝐵))
8	6, 7	mpbid 145	. . . . . . . 8 ⊢ (𝐵 ∈ On → ∪ suc 𝐵 = 𝐵)
9		suceq 4157	. . . . . . . 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 133	. . . . . 6 ⊢ (suc 𝐵 ∈ On → suc ∪ suc 𝐵 = suc 𝐵)
12		eloni 4130	. . . . . . . 8 ⊢ (suc 𝐵 ∈ On → Ord suc 𝐵)
13		ordtr 4133	. . . . . . . 8 ⊢ (Ord suc 𝐵 → Tr suc 𝐵)
14	12, 13	syl 14	. . . . . . 7 ⊢ (suc 𝐵 ∈ On → Tr suc 𝐵)
15		unisucg 4169	. . . . . . 7 ⊢ (suc 𝐵 ∈ On → (Tr suc 𝐵 ↔ ∪ suc suc 𝐵 = suc 𝐵))
16	14, 15	mpbid 145	. . . . . 6 ⊢ (suc 𝐵 ∈ On → ∪ suc suc 𝐵 = suc 𝐵)
17	11, 16	eqtr4d 2116	. . . . 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 3610	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → ∪ 𝐴 = ∪ suc 𝐵)
20		suceq 4157	. . . . . 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 4157	. . . . . 6 ⊢ (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
23	22	unieqd 3612	. . . . 5 ⊢ (𝐴 = suc 𝐵 → ∪ suc 𝐴 = ∪ suc suc 𝐵)
24	21, 23	eqeq12d 2095	. . . 4 ⊢ (𝐴 = suc 𝐵 → (suc ∪ 𝐴 = ∪ suc 𝐴 ↔ suc ∪ suc 𝐵 = ∪ suc suc 𝐵))
25	18, 24	syl5ibr 154	. . 3 ⊢ (𝐴 = suc 𝐵 → ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = ∪ suc 𝐴))
26	25	anabsi7 545	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = ∪ suc 𝐴)
27		eloni 4130	. . . . 5 ⊢ (𝐴 ∈ On → Ord 𝐴)
28		ordtr 4133	. . . . 5 ⊢ (Ord 𝐴 → Tr 𝐴)
29	27, 28	syl 14	. . . 4 ⊢ (𝐴 ∈ On → Tr 𝐴)
30		unisucg 4169	. . . 4 ⊢ (𝐴 ∈ On → (Tr 𝐴 ↔ ∪ suc 𝐴 = 𝐴))
31	29, 30	mpbid 145	. . 3 ⊢ (𝐴 ∈ On → ∪ suc 𝐴 = 𝐴)
32	31	adantr 270	. 2 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → ∪ suc 𝐴 = 𝐴)
33	26, 32	eqtrd 2113	1 ⊢ ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc ∪ 𝐴 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 102   = wceq 1284   ∈ wcel 1433  ∪ cuni 3601  Tr wtr 3875  Ord word 4117  Oncon0 4118  suc csuc 4120

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-uni 3602  df-tr 3876  df-iord 4121  df-on 4123  df-suc 4126

This theorem is referenced by: (None)