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Theorem elelsuc 4269
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 674 . 2  |-  ( A  e.  B  ->  ( A  e.  B  \/  A  =  B )
)
2 elsucg 4264 . 2  |-  ( A  e.  B  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
31, 2mpbird 166 1  |-  ( A  e.  B  ->  A  e.  suc  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 670    = wceq 1299    e. wcel 1448   suc csuc 4225
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-sn 3480  df-suc 4231
This theorem is referenced by:  suctr  4281  ordsuc  4416  nnaordex  6353  fiintim  6746  exmidfodomrlemr  6967  exmidfodomrlemrALT  6968  ennnfonelemex  11719
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