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Theorem elsuc 4888
Description: Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-2003.)
Hypothesis
Ref Expression
elsuc.1  |-  A  e. 
_V
Assertion
Ref Expression
elsuc  |-  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) )

Proof of Theorem elsuc
StepHypRef Expression
1 elsuc.1 . 2  |-  A  e. 
_V
2 elsucg 4886 . 2  |-  ( A  e.  _V  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
31, 2ax-mp 5 1  |-  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    \/ wo 368    = wceq 1370    e. wcel 1758   _Vcvv 3070   suc csuc 4821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-v 3072  df-un 3433  df-sn 3978  df-suc 4825
This theorem is referenced by:  sucel  4892  suctr  4902  limsssuc  6563  omsmolem  7194  cantnfle  7982  cantnfleOLD  8012  infxpenlem  8283  inatsk  9048  untsucf  27497  dfon2lem7  27738
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