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Theorem trsuc 4666
 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 4658 . . . . . 6
2 ssexg 4349 . . . . . 6
31, 2mpan 652 . . . . 5
4 sucidg 4659 . . . . 5
53, 4syl 16 . . . 4
65ancri 536 . . 3
7 trel 4309 . . 3
86, 7syl5 30 . 2
98imp 419 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wcel 1725  cvv 2956   wss 3320   wtr 4302   csuc 4583 This theorem is referenced by:  onuninsuci  4820  limsuc  4829  tz7.44-2  6665  cantnflt  7627  cantnfp1lem3  7636  cantnflem1b  7642  cantnflem1  7645  cnfcom  7657  axdc3lem2  8331  inar1  8650  limsuc2  27115  bnj967  29316 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-un 3325  df-in 3327  df-ss 3334  df-sn 3820  df-uni 4016  df-tr 4303  df-suc 4587
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