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Theorem elsnc2g 3669
Description: There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15. This variation requires only that  B, rather than  A, be a set. (Contributed by NM, 28-Oct-2003.)
Assertion
Ref Expression
elsnc2g  |-  ( B  e.  V  ->  ( A  e.  { B } 
<->  A  =  B ) )

Proof of Theorem elsnc2g
StepHypRef Expression
1 elsni 3665 . 2  |-  ( A  e.  { B }  ->  A  =  B )
2 snidg 3666 . . 3  |-  ( B  e.  V  ->  B  e.  { B } )
3 eleq1 2344 . . 3  |-  ( A  =  B  ->  ( A  e.  { B } 
<->  B  e.  { B } ) )
42, 3syl5ibrcom 213 . 2  |-  ( B  e.  V  ->  ( A  =  B  ->  A  e.  { B }
) )
51, 4impbid2 195 1  |-  ( B  e.  V  ->  ( A  e.  { B } 
<->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1623    e. wcel 1685   {csn 3641
This theorem is referenced by:  elsnc2  3670  elsuc2g  4459  mptiniseg  5165  fzosplitsni  10917  limcco  19239  ply1termlem  19581  stirlinglem8  27241  elpmapat  29232
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-v 2791  df-sn 3647
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