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Theorem elsnc2g 3572
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 3568 . 2  |-  ( A  e.  { B }  ->  A  =  B )
2 snidg 3569 . . 3  |-  ( B  e.  V  ->  B  e.  { B } )
3 eleq1 2313 . . 3  |-  ( A  =  B  ->  ( A  e.  { B } 
<->  B  e.  { B } ) )
42, 3syl5ibrcom 215 . 2  |-  ( B  e.  V  ->  ( A  =  B  ->  A  e.  { B }
) )
51, 4impbid2 197 1  |-  ( B  e.  V  ->  ( A  e.  { B } 
<->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    = wceq 1619    e. wcel 1621   {csn 3544
This theorem is referenced by:  elsnc2  3573  elsuc2g  4353  mptiniseg  5073  fzosplitsni  10799  limcco  19075  ply1termlem  19417  elpmapat  28642
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-v 2729  df-sn 3550
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