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Theorem elsucg 4459
Description: Membership in a successor. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 15-Sep-1995.)
Assertion
Ref Expression
elsucg  |-  ( A  e.  V  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )

Proof of Theorem elsucg
StepHypRef Expression
1 df-suc 4398 . . . 4  |-  suc  B  =  ( B  u.  { B } )
21eleq2i 2349 . . 3  |-  ( A  e.  suc  B  <->  A  e.  ( B  u.  { B } ) )
3 elun 3318 . . 3  |-  ( A  e.  ( B  u.  { B } )  <->  ( A  e.  B  \/  A  e.  { B } ) )
42, 3bitri 242 . 2  |-  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  e.  { B } ) )
5 elsncg 3664 . . 3  |-  ( A  e.  V  ->  ( A  e.  { B } 
<->  A  =  B ) )
65orbi2d 684 . 2  |-  ( A  e.  V  ->  (
( A  e.  B  \/  A  e.  { B } )  <->  ( A  e.  B  \/  A  =  B ) ) )
74, 6syl5bb 250 1  |-  ( A  e.  V  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    \/ wo 359    = wceq 1624    e. wcel 1685    u. cun 3152   {csn 3642   suc csuc 4394
This theorem is referenced by:  elsuc  4461  elelsuc  4464  sucidg  4470  ordsssuc  4479  ordsucelsuc  4613  suc11reg  7316  nlt1pi  8526
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-v 2792  df-un 3159  df-sn 3648  df-suc 4398
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