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Theorem elsuc 4224
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 4222 . 2  |-  ( A  e.  _V  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
31, 2ax-mp 7 1  |-  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 103    \/ wo 664    = wceq 1289    e. wcel 1438   _Vcvv 2619   suc csuc 4183
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3001  df-sn 3447  df-suc 4189
This theorem is referenced by:  sucel  4228  suctr  4239  0elsucexmid  4371  tfrlemisucaccv  6072  tfr1onlemsucaccv  6088  tfrcllemsucaccv  6101
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