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Theorem elsuc2 4611
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 4609 . 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 177    \/ wo 358    = wceq 1649    e. wcel 1721   _Vcvv 2916   suc csuc 4543
This theorem is referenced by:  alephordi  7911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-un 3285  df-sn 3780  df-suc 4547
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