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Theorem trsuc 4967
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 4960 . . . . . 6
2 ssexg 4598 . . . . . 6
31, 2mpan 670 . . . . 5
4 sucidg 4961 . . . . 5
53, 4syl 16 . . . 4
65ancri 552 . . 3
7 trel 4552 . . 3
86, 7syl5 32 . 2
98imp 429 1
Colors of variables: wff setvar class
Syntax hints:  ->wi 4  /\wa 369  e.wcel 1818   cvv 3109  C_wss 3475  Trwtr 4545  succsuc 4885
This theorem is referenced by:  onuninsuci  6675  limsuc  6684  tz7.44-2  7092  cantnflt  8112  cantnfp1lem3  8120  cantnflem1b  8126  cantnflem1  8129  cantnfltOLD  8142  cantnfp1lem3OLD  8146  cantnflem1bOLD  8149  cantnflem1OLD  8152  cnfcom  8165  cnfcomOLD  8173  axdc3lem2  8852  inar1  9174  limsuc2  30986  bnj967  34003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-un 3480  df-in 3482  df-ss 3489  df-sn 4030  df-uni 4250  df-tr 4546  df-suc 4889
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