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Theorem elsuci 4581
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 4521 . . . 4  |-  suc  B  =  ( B  u.  { B } )
21eleq2i 2444 . . 3  |-  ( A  e.  suc  B  <->  A  e.  ( B  u.  { B } ) )
3 elun 3424 . . 3  |-  ( A  e.  ( B  u.  { B } )  <->  ( A  e.  B  \/  A  e.  { B } ) )
42, 3bitri 241 . 2  |-  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  e.  { B } ) )
5 elsni 3774 . . 3  |-  ( A  e.  { B }  ->  A  =  B )
65orim2i 505 . 2  |-  ( ( A  e.  B  \/  A  e.  { B } )  ->  ( A  e.  B  \/  A  =  B )
)
74, 6sylbi 188 1  |-  ( A  e.  suc  B  -> 
( A  e.  B  \/  A  =  B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 358    = wceq 1649    e. wcel 1717    u. cun 3254   {csn 3750   suc csuc 4517
This theorem is referenced by:  trsucss  4600  ordnbtwn  4605  suc11  4618  tfrlem11  6578  omordi  6738  nnmordi  6803  phplem3  7217  pssnn  7256  r1sdom  7626  cfsuc  8063  axdc3lem2  8257  axdc3lem4  8259  indpi  8710  ontgval  25888  onsucconi  25894  suctrALT2VD  28282  suctrALT2  28283  suctrALTcf  28368  suctrALTcfVD  28369  suctrALT3  28370  bnj563  28442  bnj964  28645
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-v 2894  df-un 3261  df-sn 3756  df-suc 4521
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