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Theorem elsuc2 4541
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 4539 . 2  |-  ( A  e.  _V  ->  ( B  e.  suc  A  <->  ( B  e.  A  \/  B  =  A ) ) )
31, 2ax-mp 8 1  |-  ( B  e.  suc  A  <->  ( B  e.  A  \/  B  =  A ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    \/ wo 357    = wceq 1642    e. wcel 1710   _Vcvv 2864   suc csuc 4473
This theorem is referenced by:  alephordi  7788
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-v 2866  df-un 3233  df-sn 3722  df-suc 4477
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