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Theorem snelsingles 25485
Description: A singleton is a member of the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
Hypothesis
Ref Expression
snelsingles.1  |-  A  e. 
_V
Assertion
Ref Expression
snelsingles  |-  { A }  e.  Singletons

Proof of Theorem snelsingles
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 snelsingles.1 . . . 4  |-  A  e. 
_V
2 isset 2903 . . . . 5  |-  ( A  e.  _V  <->  E. x  x  =  A )
3 eqcom 2389 . . . . . 6  |-  ( x  =  A  <->  A  =  x )
43exbii 1589 . . . . 5  |-  ( E. x  x  =  A  <->  E. x  A  =  x )
52, 4bitri 241 . . . 4  |-  ( A  e.  _V  <->  E. x  A  =  x )
61, 5mpbi 200 . . 3  |-  E. x  A  =  x
7 sneq 3768 . . 3  |-  ( A  =  x  ->  { A }  =  { x } )
86, 7eximii 1584 . 2  |-  E. x { A }  =  {
x }
9 elsingles 25481 . 2  |-  ( { A }  e.  Singletons  <->  E. x { A }  =  { x } )
108, 9mpbir 201 1  |-  { A }  e.  Singletons
Colors of variables: wff set class
Syntax hints:   E.wex 1547    = wceq 1649    e. wcel 1717   _Vcvv 2899   {csn 3757   Singletonscsingles 25406
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-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-eprel 4435  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-fo 5400  df-fv 5402  df-1st 6288  df-2nd 6289  df-symdif 25386  df-txp 25419  df-singleton 25427  df-singles 25428
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