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Theorem elsuc2g 4563
Description: Variant of membership in a successor, requiring that  B rather than  A be a set. (Contributed by NM, 28-Oct-2003.)
Assertion
Ref Expression
elsuc2g  |-  ( B  e.  V  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )

Proof of Theorem elsuc2g
StepHypRef Expression
1 df-suc 4501 . . 3  |-  suc  B  =  ( B  u.  { B } )
21eleq2i 2430 . 2  |-  ( A  e.  suc  B  <->  A  e.  ( B  u.  { B } ) )
3 elun 3404 . . 3  |-  ( A  e.  ( B  u.  { B } )  <->  ( A  e.  B  \/  A  e.  { B } ) )
4 elsnc2g 3757 . . . 4  |-  ( B  e.  V  ->  ( A  e.  { B } 
<->  A  =  B ) )
54orbi2d 682 . . 3  |-  ( B  e.  V  ->  (
( A  e.  B  \/  A  e.  { B } )  <->  ( A  e.  B  \/  A  =  B ) ) )
63, 5syl5bb 248 . 2  |-  ( B  e.  V  ->  ( A  e.  ( B  u.  { B } )  <-> 
( A  e.  B  \/  A  =  B
) ) )
72, 6syl5bb 248 1  |-  ( B  e.  V  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    \/ wo 357    = wceq 1647    e. wcel 1715    u. cun 3236   {csn 3729   suc csuc 4497
This theorem is referenced by:  elsuc2  4565  om2uzlti  11177
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-v 2875  df-un 3243  df-sn 3735  df-suc 4501
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