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Theorem elsuci 4474
Description: Membership in a successor. This one-way implication does not require that either  A or  B be sets. (Contributed by NM, 6-Jun-1994.)
Assertion
Ref Expression
elsuci  |-  ( A  e.  suc  B  -> 
( A  e.  B  \/  A  =  B
) )

Proof of Theorem elsuci
StepHypRef Expression
1 df-suc 4414 . . . 4  |-  suc  B  =  ( B  u.  { B } )
21eleq2i 2360 . . 3  |-  ( A  e.  suc  B  <->  A  e.  ( B  u.  { B } ) )
3 elun 3329 . . 3  |-  ( A  e.  ( B  u.  { B } )  <->  ( A  e.  B  \/  A  e.  { B } ) )
42, 3bitri 240 . 2  |-  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  e.  { B } ) )
5 elsni 3677 . . 3  |-  ( A  e.  { B }  ->  A  =  B )
65orim2i 504 . 2  |-  ( ( A  e.  B  \/  A  e.  { B } )  ->  ( A  e.  B  \/  A  =  B )
)
74, 6sylbi 187 1  |-  ( A  e.  suc  B  -> 
( A  e.  B  \/  A  =  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 357    = wceq 1632    e. wcel 1696    u. cun 3163   {csn 3653   suc csuc 4410
This theorem is referenced by:  trsucss  4494  ordnbtwn  4499  suc11  4512  tfrlem11  6420  omordi  6580  nnmordi  6645  phplem3  7058  pssnn  7097  r1sdom  7462  cfsuc  7899  axdc3lem2  8093  axdc3lem4  8095  indpi  8547  ontgval  24942  onsucconi  24948  suctrALT2VD  28928  suctrALT2  28929  suctrALTcf  29014  suctrALTcfVD  29015  suctrALT3  29016  suctrALT4  29020  bnj563  29088  bnj964  29291
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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