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Theorem elsuc2g 4652
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 4590 . . 3  |-  suc  B  =  ( B  u.  { B } )
21eleq2i 2502 . 2  |-  ( A  e.  suc  B  <->  A  e.  ( B  u.  { B } ) )
3 elun 3490 . . 3  |-  ( A  e.  ( B  u.  { B } )  <->  ( A  e.  B  \/  A  e.  { B } ) )
4 elsnc2g 3844 . . . 4  |-  ( B  e.  V  ->  ( A  e.  { B } 
<->  A  =  B ) )
54orbi2d 684 . . 3  |-  ( B  e.  V  ->  (
( A  e.  B  \/  A  e.  { B } )  <->  ( A  e.  B  \/  A  =  B ) ) )
63, 5syl5bb 250 . 2  |-  ( B  e.  V  ->  ( A  e.  ( B  u.  { B } )  <-> 
( A  e.  B  \/  A  =  B
) ) )
72, 6syl5bb 250 1  |-  ( B  e.  V  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    \/ wo 359    = wceq 1653    e. wcel 1726    u. cun 3320   {csn 3816   suc csuc 4586
This theorem is referenced by:  elsuc2  4654  om2uzlti  11295
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-v 2960  df-un 3327  df-sn 3822  df-suc 4590
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