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Theorem elsuc2 5512
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 5510 . 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 setvar class
Syntax hints:    <-> wb 187    \/ wo 369    = wceq 1437    e. wcel 1870   _Vcvv 3087   suc csuc 5444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-v 3089  df-un 3447  df-sn 4003  df-suc 5448
This theorem is referenced by:  alephordi  8503
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