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Theorem elsnc2g 3787
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 3783 . 2  |-  ( A  e.  { B }  ->  A  =  B )
2 snidg 3784 . . 3  |-  ( B  e.  V  ->  B  e.  { B } )
3 eleq1 2449 . . 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 1649    e. wcel 1717   {csn 3759
This theorem is referenced by:  elsnc2  3788  elsuc2g  4592  mptiniseg  5306  fzosplitsni  11125  limcco  19649  ply1termlem  19991  stirlinglem8  27500  elpmapat  29880
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
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 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-v 2903  df-sn 3765
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