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Theorem elsuc2 4324
Description: Membership in a successor. (Contributed by NM, 15-Sep-2003.)
Hypothesis
Ref Expression
elsuc.1  |-  A  e. 
_V
Assertion
Ref Expression
elsuc2  |-  ( B  e.  suc  A  <->  ( B  e.  A  \/  B  =  A ) )

Proof of Theorem elsuc2
StepHypRef Expression
1 elsuc.1 . 2  |-  A  e. 
_V
2 elsuc2g 4322 . 2  |-  ( A  e.  _V  ->  ( B  e.  suc  A  <->  ( B  e.  A  \/  B  =  A ) ) )
31, 2ax-mp 5 1  |-  ( B  e.  suc  A  <->  ( B  e.  A  \/  B  =  A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    \/ wo 697    = wceq 1331    e. wcel 1480   _Vcvv 2681   suc csuc 4282
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-un 3070  df-sn 3528  df-suc 4288
This theorem is referenced by:  nnsucelsuc  6380
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