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Theorem elelsuc 4480
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 4475 . 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 1632    e. wcel 1696   suc csuc 4410
This theorem is referenced by:  suctr  4491  pssnn  7097  pwsdompw  7846  fin1a2lem4  8045  grur1a  8457  bnj570  29253
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-sn 3659  df-suc 4414
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