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Theorem elsucg 4264
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 4231 . . . 4  |-  suc  B  =  ( B  u.  { B } )
21eleq2i 2166 . . 3  |-  ( A  e.  suc  B  <->  A  e.  ( B  u.  { B } ) )
3 elun 3164 . . 3  |-  ( A  e.  ( B  u.  { B } )  <->  ( A  e.  B  \/  A  e.  { B } ) )
42, 3bitri 183 . 2  |-  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  e.  { B } ) )
5 elsng 3489 . . 3  |-  ( A  e.  V  ->  ( A  e.  { B } 
<->  A  =  B ) )
65orbi2d 745 . 2  |-  ( A  e.  V  ->  (
( A  e.  B  \/  A  e.  { B } )  <->  ( A  e.  B  \/  A  =  B ) ) )
74, 6syl5bb 191 1  |-  ( A  e.  V  ->  ( A  e.  suc  B  <->  ( A  e.  B  \/  A  =  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    \/ wo 670    = wceq 1299    e. wcel 1448    u. cun 3019   {csn 3474   suc csuc 4225
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-sn 3480  df-suc 4231
This theorem is referenced by:  elsuc  4266  elelsuc  4269  sucidg  4276  onsucelsucr  4362  onsucsssucexmid  4380  suc11g  4410  nnsssuc  6328  nlt1pig  7050  bj-peano4  12738
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