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Theorem elelsuc 4427
Description: Membership in a successor. (Contributed by NM, 20-Jun-1998.)
Assertion
Ref Expression
elelsuc  |-  ( A  e.  B  ->  A  e.  suc  B )

Proof of Theorem elelsuc
StepHypRef Expression
1 orc 713 . 2  |-  ( A  e.  B  ->  ( A  e.  B  \/  A  =  B )
)
2 elsucg 4422 . 2  |-  ( A  e.  B  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
31, 2mpbird 167 1  |-  ( A  e.  B  ->  A  e.  suc  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 709    = wceq 1364    e. wcel 2160   suc csuc 4383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-un 3148  df-sn 3613  df-suc 4389
This theorem is referenced by:  suctr  4439  ordsuc  4580  nnaordex  6553  fiintim  6957  exmidfodomrlemr  7231  exmidfodomrlemrALT  7232  3nelsucpw1  7263  ennnfonelemex  12465
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