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Theorem elsuc 4936
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 4934 . 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 366    = wceq 1398    e. wcel 1823   _Vcvv 3106   suc csuc 4869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-un 3466  df-sn 4017  df-suc 4873
This theorem is referenced by:  sucel  4940  suctr  4950  limsssuc  6658  omsmolem  7294  cantnfle  8081  cantnfleOLD  8111  infxpenlem  8382  inatsk  9145  untsucf  29323  dfon2lem7  29461
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