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Theorem elsucg 5508
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 5447 . . . 4  |-  suc  B  =  ( B  u.  { B } )
21eleq2i 2531 . . 3  |-  ( A  e.  suc  B  <->  A  e.  ( B  u.  { B } ) )
3 elun 3585 . . 3  |-  ( A  e.  ( B  u.  { B } )  <->  ( A  e.  B  \/  A  e.  { B } ) )
42, 3bitri 257 . 2  |-  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  e.  { B } ) )
5 elsncg 4002 . . 3  |-  ( A  e.  V  ->  ( A  e.  { B } 
<->  A  =  B ) )
65orbi2d 713 . 2  |-  ( A  e.  V  ->  (
( A  e.  B  \/  A  e.  { B } )  <->  ( A  e.  B  \/  A  =  B ) ) )
74, 6syl5bb 265 1  |-  ( A  e.  V  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ wo 374    = wceq 1454    e. wcel 1897    u. cun 3413   {csn 3979   suc csuc 5443
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-v 3058  df-un 3420  df-sn 3980  df-suc 5447
This theorem is referenced by:  elsuc  5510  elelsuc  5513  sucidg  5519  ordsssuc  5527  ordsucelsuc  6675  suc11reg  8149  nlt1pi  9356
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