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Theorem onsucuni2 4564
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 2240 . . . . . 6 (𝐴 = suc 𝐵 → (𝐴 ∈ On ↔ suc 𝐵 ∈ On))
21biimpac 298 . . . . 5 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐵 ∈ On)
3 onsucb 4503 . . . . . . 7 (𝐵 ∈ On ↔ suc 𝐵 ∈ On)
4 eloni 4376 . . . . . . . . . 10 (𝐵 ∈ On → Ord 𝐵)
5 ordtr 4379 . . . . . . . . . 10 (Ord 𝐵 → Tr 𝐵)
64, 5syl 14 . . . . . . . . 9 (𝐵 ∈ On → Tr 𝐵)
7 unisucg 4415 . . . . . . . . 9 (𝐵 ∈ On → (Tr 𝐵 suc 𝐵 = 𝐵))
86, 7mpbid 147 . . . . . . . 8 (𝐵 ∈ On → suc 𝐵 = 𝐵)
9 suceq 4403 . . . . . . . 8 ( suc 𝐵 = 𝐵 → suc suc 𝐵 = suc 𝐵)
108, 9syl 14 . . . . . . 7 (𝐵 ∈ On → suc suc 𝐵 = suc 𝐵)
113, 10sylbir 135 . . . . . 6 (suc 𝐵 ∈ On → suc suc 𝐵 = suc 𝐵)
12 eloni 4376 . . . . . . . 8 (suc 𝐵 ∈ On → Ord suc 𝐵)
13 ordtr 4379 . . . . . . . 8 (Ord suc 𝐵 → Tr suc 𝐵)
1412, 13syl 14 . . . . . . 7 (suc 𝐵 ∈ On → Tr suc 𝐵)
15 unisucg 4415 . . . . . . 7 (suc 𝐵 ∈ On → (Tr suc 𝐵 suc suc 𝐵 = suc 𝐵))
1614, 15mpbid 147 . . . . . 6 (suc 𝐵 ∈ On → suc suc 𝐵 = suc 𝐵)
1711, 16eqtr4d 2213 . . . . 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 3819 . . . . . 6 (𝐴 = suc 𝐵 𝐴 = suc 𝐵)
20 suceq 4403 . . . . . 6 ( 𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
2119, 20syl 14 . . . . 5 (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
22 suceq 4403 . . . . . 6 (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
2322unieqd 3821 . . . . 5 (𝐴 = suc 𝐵 suc 𝐴 = suc suc 𝐵)
2421, 23eqeq12d 2192 . . . 4 (𝐴 = suc 𝐵 → (suc 𝐴 = suc 𝐴 ↔ suc suc 𝐵 = suc suc 𝐵))
2518, 24imbitrrid 156 . . 3 (𝐴 = suc 𝐵 → ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = suc 𝐴))
2625anabsi7 581 . 2 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = suc 𝐴)
27 eloni 4376 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
28 ordtr 4379 . . . . 5 (Ord 𝐴 → Tr 𝐴)
2927, 28syl 14 . . . 4 (𝐴 ∈ On → Tr 𝐴)
30 unisucg 4415 . . . 4 (𝐴 ∈ On → (Tr 𝐴 suc 𝐴 = 𝐴))
3129, 30mpbid 147 . . 3 (𝐴 ∈ On → suc 𝐴 = 𝐴)
3231adantr 276 . 2 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = 𝐴)
3326, 32eqtrd 2210 1 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wcel 2148   cuni 3810  Tr wtr 4102  Ord word 4363  Oncon0 4364  suc csuc 4366
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-uni 3811  df-tr 4103  df-iord 4367  df-on 4369  df-suc 4372
This theorem is referenced by:  nnsucpred  4617  nnpredcl  4623
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