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Theorem elsucg 4612
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 4551 . . . 4  |-  suc  B  =  ( B  u.  { B } )
21eleq2i 2472 . . 3  |-  ( A  e.  suc  B  <->  A  e.  ( B  u.  { B } ) )
3 elun 3452 . . 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 elsncg 3800 . . 3  |-  ( A  e.  V  ->  ( A  e.  { B } 
<->  A  =  B ) )
65orbi2d 683 . 2  |-  ( A  e.  V  ->  (
( A  e.  B  \/  A  e.  { B } )  <->  ( A  e.  B  \/  A  =  B ) ) )
74, 6syl5bb 249 1  |-  ( A  e.  V  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    \/ wo 358    = wceq 1649    e. wcel 1721    u. cun 3282   {csn 3778   suc csuc 4547
This theorem is referenced by:  elsuc  4614  elelsuc  4617  sucidg  4623  ordsssuc  4631  ordsucelsuc  4765  suc11reg  7534  nlt1pi  8743
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
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 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-v 2922  df-un 3289  df-sn 3784  df-suc 4551
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