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Theorem elsncg 3675
Description: There is only one element in a singleton. Exercise 2 of [TakeutiZaring] p. 15 (generalized). (Contributed by NM, 13-Sep-1995.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
elsncg  |-  ( A  e.  V  ->  ( A  e.  { B } 
<->  A  =  B ) )

Proof of Theorem elsncg
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2302 . 2  |-  ( x  =  A  ->  (
x  =  B  <->  A  =  B ) )
2 df-sn 3659 . 2  |-  { B }  =  { x  |  x  =  B }
31, 2elab2g 2929 1  |-  ( A  e.  V  ->  ( A  e.  { B } 
<->  A  =  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1632    e. wcel 1696   {csn 3653
This theorem is referenced by:  elsnc  3676  elsni  3677  snidg  3678  eltpg  3689  eldifsn  3762  elsucg  4475  ltxr  10473  elfzp12  10877  ramcl  13092  xrge0tsmsbi  23398  itgaddnclem2  25010  lineval5a  26191  lineval6a  26192  nbcusgra  28298
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-sn 3659
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