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Theorem elsnc2g 3834
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 3830 . 2  |-  ( A  e.  { B }  ->  A  =  B )
2 snidg 3831 . . 3  |-  ( B  e.  V  ->  B  e.  { B } )
3 eleq1 2495 . . 3  |-  ( A  =  B  ->  ( A  e.  { B } 
<->  B  e.  { B } ) )
42, 3syl5ibrcom 214 . 2  |-  ( B  e.  V  ->  ( A  =  B  ->  A  e.  { B }
) )
51, 4impbid2 196 1  |-  ( B  e.  V  ->  ( A  e.  { B } 
<->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652    e. wcel 1725   {csn 3806
This theorem is referenced by:  elsnc2  3835  elsuc2g  4641  mptiniseg  5356  fzosplitsni  11188  limcco  19772  ply1termlem  20114  stirlinglem8  27797  elpmapat  30498
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-sn 3812
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