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Theorem onsucuni2 4288
 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 2100 . . . . . 6 (𝐴 = suc 𝐵 → (𝐴 ∈ On ↔ suc 𝐵 ∈ On))
21biimpac 282 . . . . 5 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐵 ∈ On)
3 sucelon 4229 . . . . . . 7 (𝐵 ∈ On ↔ suc 𝐵 ∈ On)
4 eloni 4112 . . . . . . . . . 10 (𝐵 ∈ On → Ord 𝐵)
5 ordtr 4115 . . . . . . . . . 10 (Ord 𝐵 → Tr 𝐵)
64, 5syl 14 . . . . . . . . 9 (𝐵 ∈ On → Tr 𝐵)
7 unisucg 4151 . . . . . . . . 9 (𝐵 ∈ On → (Tr 𝐵 suc 𝐵 = 𝐵))
86, 7mpbid 135 . . . . . . . 8 (𝐵 ∈ On → suc 𝐵 = 𝐵)
9 suceq 4139 . . . . . . . 8 ( suc 𝐵 = 𝐵 → suc suc 𝐵 = suc 𝐵)
108, 9syl 14 . . . . . . 7 (𝐵 ∈ On → suc suc 𝐵 = suc 𝐵)
113, 10sylbir 125 . . . . . 6 (suc 𝐵 ∈ On → suc suc 𝐵 = suc 𝐵)
12 eloni 4112 . . . . . . . 8 (suc 𝐵 ∈ On → Ord suc 𝐵)
13 ordtr 4115 . . . . . . . 8 (Ord suc 𝐵 → Tr suc 𝐵)
1412, 13syl 14 . . . . . . 7 (suc 𝐵 ∈ On → Tr suc 𝐵)
15 unisucg 4151 . . . . . . 7 (suc 𝐵 ∈ On → (Tr suc 𝐵 suc suc 𝐵 = suc 𝐵))
1614, 15mpbid 135 . . . . . 6 (suc 𝐵 ∈ On → suc suc 𝐵 = suc 𝐵)
1711, 16eqtr4d 2075 . . . . 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 3589 . . . . . 6 (𝐴 = suc 𝐵 𝐴 = suc 𝐵)
20 suceq 4139 . . . . . 6 ( 𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
2119, 20syl 14 . . . . 5 (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
22 suceq 4139 . . . . . 6 (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
2322unieqd 3591 . . . . 5 (𝐴 = suc 𝐵 suc 𝐴 = suc suc 𝐵)
2421, 23eqeq12d 2054 . . . 4 (𝐴 = suc 𝐵 → (suc 𝐴 = suc 𝐴 ↔ suc suc 𝐵 = suc suc 𝐵))
2518, 24syl5ibr 145 . . 3 (𝐴 = suc 𝐵 → ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = suc 𝐴))
2625anabsi7 515 . 2 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = suc 𝐴)
27 eloni 4112 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
28 ordtr 4115 . . . . 5 (Ord 𝐴 → Tr 𝐴)
2927, 28syl 14 . . . 4 (𝐴 ∈ On → Tr 𝐴)
30 unisucg 4151 . . . 4 (𝐴 ∈ On → (Tr 𝐴 suc 𝐴 = 𝐴))
3129, 30mpbid 135 . . 3 (𝐴 ∈ On → suc 𝐴 = 𝐴)
3231adantr 261 . 2 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = 𝐴)
3326, 32eqtrd 2072 1 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = 𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243   ∈ wcel 1393  ∪ cuni 3580  Tr wtr 3854  Ord word 4099  Oncon0 4100  suc csuc 4102 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108 This theorem is referenced by: (None)
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