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Theorem elsuc 4463
Description: Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.)
Hypothesis
Ref Expression
elsuc.1  |-  A  e. 
_V
Assertion
Ref Expression
elsuc  |-  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) )

Proof of Theorem elsuc
StepHypRef Expression
1 elsuc.1 . 2  |-  A  e. 
_V
2 elsucg 4461 . 2  |-  ( A  e.  _V  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
31, 2ax-mp 8 1  |-  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    = wceq 1625    e. wcel 1686   _Vcvv 2790   suc csuc 4396
This theorem is referenced by:  sucel  4467  suctr  4477  limsssuc  4643  omsmolem  6653  cantnfle  7374  infxpenlem  7643  inatsk  8402  untsucf  24058  dfon2lem7  24147
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-v 2792  df-un 3159  df-sn 3648  df-suc 4400
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