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Theorem onsucuni2 4370
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 2150 . . . . . 6 (𝐴 = suc 𝐵 → (𝐴 ∈ On ↔ suc 𝐵 ∈ On))
21biimpac 292 . . . . 5 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐵 ∈ On)
3 sucelon 4310 . . . . . . 7 (𝐵 ∈ On ↔ suc 𝐵 ∈ On)
4 eloni 4193 . . . . . . . . . 10 (𝐵 ∈ On → Ord 𝐵)
5 ordtr 4196 . . . . . . . . . 10 (Ord 𝐵 → Tr 𝐵)
64, 5syl 14 . . . . . . . . 9 (𝐵 ∈ On → Tr 𝐵)
7 unisucg 4232 . . . . . . . . 9 (𝐵 ∈ On → (Tr 𝐵 suc 𝐵 = 𝐵))
86, 7mpbid 145 . . . . . . . 8 (𝐵 ∈ On → suc 𝐵 = 𝐵)
9 suceq 4220 . . . . . . . 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 4193 . . . . . . . 8 (suc 𝐵 ∈ On → Ord suc 𝐵)
13 ordtr 4196 . . . . . . . 8 (Ord suc 𝐵 → Tr suc 𝐵)
1412, 13syl 14 . . . . . . 7 (suc 𝐵 ∈ On → Tr suc 𝐵)
15 unisucg 4232 . . . . . . 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 2123 . . . . 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 3657 . . . . . 6 (𝐴 = suc 𝐵 𝐴 = suc 𝐵)
20 suceq 4220 . . . . . 6 ( 𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
2119, 20syl 14 . . . . 5 (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
22 suceq 4220 . . . . . 6 (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
2322unieqd 3659 . . . . 5 (𝐴 = suc 𝐵 suc 𝐴 = suc suc 𝐵)
2421, 23eqeq12d 2102 . . . 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 548 . 2 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = suc 𝐴)
27 eloni 4193 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
28 ordtr 4196 . . . . 5 (Ord 𝐴 → Tr 𝐴)
2927, 28syl 14 . . . 4 (𝐴 ∈ On → Tr 𝐴)
30 unisucg 4232 . . . 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 2120 1 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1289  wcel 1438   cuni 3648  Tr wtr 3928  Ord word 4180  Oncon0 4181  suc csuc 4183
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-uni 3649  df-tr 3929  df-iord 4184  df-on 4186  df-suc 4189
This theorem is referenced by:  nnpredcl  11547  nnsucpred  11548
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