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Theorem elelsuc 4645
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 375 . 2  |-  ( A  e.  B  ->  ( A  e.  B  \/  A  =  B )
)
2 elsucg 4640 . 2  |-  ( A  e.  B  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
31, 2mpbird 224 1  |-  ( A  e.  B  ->  A  e.  suc  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    = wceq 1652    e. wcel 1725   suc csuc 4575
This theorem is referenced by:  suctrALT  4656  pssnn  7319  pwsdompw  8076  fin1a2lem4  8275  grur1a  8686  bnj570  29213
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-un 3317  df-sn 3812  df-suc 4579
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