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Theorem elelsuc 4464
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 374 . 2  |-  ( A  e.  B  ->  ( A  e.  B  \/  A  =  B )
)
2 elsucg 4459 . 2  |-  ( A  e.  B  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
31, 2mpbird 223 1  |-  ( A  e.  B  ->  A  e.  suc  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    = wceq 1623    e. wcel 1684   suc csuc 4394
This theorem is referenced by:  suctr  4475  pssnn  7081  pwsdompw  7830  fin1a2lem4  8029  grur1a  8441  bnj570  28937
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-sn 3646  df-suc 4398
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