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Theorem elsuc 4460
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 4458 . 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 1623    e. wcel 1685   _Vcvv 2789   suc csuc 4393
This theorem is referenced by:  sucel  4464  suctr  4474  limsssuc  4640  omsmolem  6647  cantnfle  7368  infxpenlem  7637  inatsk  8396  untsucf  23463  dfon2lem7  23549
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-v 2791  df-un 3158  df-sn 3647  df-suc 4397
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