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Theorem onsucuni2 4315
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 2116 . . . . . 6 (𝐴 = suc 𝐵 → (𝐴 ∈ On ↔ suc 𝐵 ∈ On))
21biimpac 286 . . . . 5 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐵 ∈ On)
3 sucelon 4256 . . . . . . 7 (𝐵 ∈ On ↔ suc 𝐵 ∈ On)
4 eloni 4139 . . . . . . . . . 10 (𝐵 ∈ On → Ord 𝐵)
5 ordtr 4142 . . . . . . . . . 10 (Ord 𝐵 → Tr 𝐵)
64, 5syl 14 . . . . . . . . 9 (𝐵 ∈ On → Tr 𝐵)
7 unisucg 4178 . . . . . . . . 9 (𝐵 ∈ On → (Tr 𝐵 suc 𝐵 = 𝐵))
86, 7mpbid 139 . . . . . . . 8 (𝐵 ∈ On → suc 𝐵 = 𝐵)
9 suceq 4166 . . . . . . . 8 ( suc 𝐵 = 𝐵 → suc suc 𝐵 = suc 𝐵)
108, 9syl 14 . . . . . . 7 (𝐵 ∈ On → suc suc 𝐵 = suc 𝐵)
113, 10sylbir 129 . . . . . 6 (suc 𝐵 ∈ On → suc suc 𝐵 = suc 𝐵)
12 eloni 4139 . . . . . . . 8 (suc 𝐵 ∈ On → Ord suc 𝐵)
13 ordtr 4142 . . . . . . . 8 (Ord suc 𝐵 → Tr suc 𝐵)
1412, 13syl 14 . . . . . . 7 (suc 𝐵 ∈ On → Tr suc 𝐵)
15 unisucg 4178 . . . . . . 7 (suc 𝐵 ∈ On → (Tr suc 𝐵 suc suc 𝐵 = suc 𝐵))
1614, 15mpbid 139 . . . . . 6 (suc 𝐵 ∈ On → suc suc 𝐵 = suc 𝐵)
1711, 16eqtr4d 2091 . . . . 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 3616 . . . . . 6 (𝐴 = suc 𝐵 𝐴 = suc 𝐵)
20 suceq 4166 . . . . . 6 ( 𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
2119, 20syl 14 . . . . 5 (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
22 suceq 4166 . . . . . 6 (𝐴 = suc 𝐵 → suc 𝐴 = suc suc 𝐵)
2322unieqd 3618 . . . . 5 (𝐴 = suc 𝐵 suc 𝐴 = suc suc 𝐵)
2421, 23eqeq12d 2070 . . . 4 (𝐴 = suc 𝐵 → (suc 𝐴 = suc 𝐴 ↔ suc suc 𝐵 = suc suc 𝐵))
2518, 24syl5ibr 149 . . 3 (𝐴 = suc 𝐵 → ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = suc 𝐴))
2625anabsi7 523 . 2 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = suc 𝐴)
27 eloni 4139 . . . . 5 (𝐴 ∈ On → Ord 𝐴)
28 ordtr 4142 . . . . 5 (Ord 𝐴 → Tr 𝐴)
2927, 28syl 14 . . . 4 (𝐴 ∈ On → Tr 𝐴)
30 unisucg 4178 . . . 4 (𝐴 ∈ On → (Tr 𝐴 suc 𝐴 = 𝐴))
3129, 30mpbid 139 . . 3 (𝐴 ∈ On → suc 𝐴 = 𝐴)
3231adantr 265 . 2 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = 𝐴)
3326, 32eqtrd 2088 1 ((𝐴 ∈ On ∧ 𝐴 = suc 𝐵) → suc 𝐴 = 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409   cuni 3607  Tr wtr 3881  Ord word 4126  Oncon0 4127  suc csuc 4129
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-uni 3608  df-tr 3882  df-iord 4130  df-on 4132  df-suc 4135
This theorem is referenced by: (None)
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