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Theorem onsucuni2 4343
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
StepHypRef Expression
1 eleq1 2145 . . . . . 6 (𝐴 = suc 𝐵 → (𝐴 ∈ On ↔ suc 𝐵 ∈ On))
21biimpac 292 . . . . 5 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐵 ∈ On)
3 sucelon 4283 . . . . . . 7 (𝐵 ∈ On ↔ suc 𝐵 ∈ On)
4 eloni 4166 . . . . . . . . . 10 (𝐵 ∈ On → Ord 𝐵)
5 ordtr 4169 . . . . . . . . . 10 (Ord 𝐵 → Tr 𝐵)
64, 5syl 14 . . . . . . . . 9 (𝐵 ∈ On → Tr 𝐵)
7 unisucg 4205 . . . . . . . . 9 (𝐵 ∈ On → (Tr 𝐵 suc 𝐵 = 𝐵))
86, 7mpbid 145 . . . . . . . 8 (𝐵 ∈ On → suc 𝐵 = 𝐵)
9 suceq 4193 . . . . . . . 8 ( suc 𝐵 = 𝐵 → suc suc 𝐵 = suc 𝐵)
108, 9syl 14 . . . . . . 7 (𝐵 ∈ On → suc suc 𝐵 = suc 𝐵)
113, 10sylbir 133 . . . . . 6 (suc 𝐵 ∈ On → suc suc 𝐵 = suc 𝐵)
12 eloni 4166 . . . . . . . 8 (suc 𝐵 ∈ On → Ord suc 𝐵)
13 ordtr 4169 . . . . . . . 8 (Ord suc 𝐵 → Tr suc 𝐵)
1412, 13syl 14 . . . . . . 7 (suc 𝐵 ∈ On → Tr suc 𝐵)
15 unisucg 4205 . . . . . . 7 (suc 𝐵 ∈ On → (Tr suc 𝐵 suc suc 𝐵 = suc 𝐵))
1614, 15mpbid 145 . . . . . 6 (suc 𝐵 ∈ On → suc suc 𝐵 = suc 𝐵)
1711, 16eqtr4d 2118 . . . . 5 (suc 𝐵 ∈ On → suc suc 𝐵 = suc suc 𝐵)
182, 17syl 14 . . . 4 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc suc 𝐵 = suc suc 𝐵)
19 unieq 3636 . . . . . 6 (𝐴 = suc 𝐵 𝐴 = suc 𝐵)
20 suceq 4193 . . . . . 6 ( 𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
2119, 20syl 14 . . . . 5 (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
22 suceq 4193 . . . . . 6 (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
2322unieqd 3638 . . . . 5 (𝐴 = suc 𝐵 suc 𝐴 = suc suc 𝐵)
2421, 23eqeq12d 2097 . . . 4 (𝐴 = suc 𝐵 → (suc 𝐴 = suc 𝐴 ↔ suc suc 𝐵 = suc suc 𝐵))
2518, 24syl5ibr 154 . . 3 (𝐴 = suc 𝐵 → ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = suc 𝐴))
2625anabsi7 546 . 2 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = suc 𝐴)
27 eloni 4166 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
28 ordtr 4169 . . . . 5 (Ord 𝐴 → Tr 𝐴)
2927, 28syl 14 . . . 4 (𝐴 ∈ On → Tr 𝐴)
30 unisucg 4205 . . . 4 (𝐴 ∈ On → (Tr 𝐴 suc 𝐴 = 𝐴))
3129, 30mpbid 145 . . 3 (𝐴 ∈ On → suc 𝐴 = 𝐴)
3231adantr 270 . 2 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = 𝐴)
3326, 32eqtrd 2115 1 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wcel 1434   cuni 3627  Tr wtr 3901  Ord word 4153  Oncon0 4154  suc csuc 4156
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-un 4224
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-uni 3628  df-tr 3902  df-iord 4157  df-on 4159  df-suc 4162
This theorem is referenced by: (None)
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