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Theorem elsuc2 4654
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 4652 . 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 178    \/ wo 359    = wceq 1653    e. wcel 1726   _Vcvv 2958   suc csuc 4586
This theorem is referenced by:  alephordi  7960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-un 3327  df-sn 3822  df-suc 4590
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