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Theorem elsucg 4141
 Description: Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.)
Assertion
Ref Expression
elsucg

Proof of Theorem elsucg
StepHypRef Expression
1 df-suc 4108 . . . 4
21eleq2i 2104 . . 3
3 elun 3084 . . 3
42, 3bitri 173 . 2
5 elsng 3390 . . 3
65orbi2d 704 . 2
74, 6syl5bb 181 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 98   wo 629   wceq 1243   wcel 1393   cun 2915  csn 3375   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 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-sn 3381  df-suc 4108 This theorem is referenced by:  elsuc  4143  elelsuc  4146  sucidg  4153  onsucelsucr  4234  onsucsssucexmid  4252  suc11g  4281  nlt1pig  6439  bj-peano4  10080
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