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Theorem trsuc 4159
 Description: A set whose successor belongs to a transitive class also belongs. (Contributed by NM, 5-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
Assertion
Ref Expression
trsuc

Proof of Theorem trsuc
StepHypRef Expression
1 sssucid 4152 . . . . . 6
2 ssexg 3896 . . . . . 6
31, 2mpan 400 . . . . 5
4 sucidg 4153 . . . . 5
53, 4syl 14 . . . 4
65ancri 307 . . 3
7 trel 3861 . . 3
86, 7syl5 28 . 2
98imp 115 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 97   wcel 1393  cvv 2557   wss 2917   wtr 3854   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-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-uni 3581  df-tr 3855  df-suc 4108 This theorem is referenced by: (None)
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