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Theorem snelsingles 23872
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 2792 . . . . 5  |-  ( A  e.  _V  <->  E. x  x  =  A )
3 eqcom 2285 . . . . . 6  |-  ( x  =  A  <->  A  =  x )
43exbii 1569 . . . . 5  |-  ( E. x  x  =  A  <->  E. x  A  =  x )
52, 4bitri 240 . . . 4  |-  ( A  e.  _V  <->  E. x  A  =  x )
61, 5mpbi 199 . . 3  |-  E. x  A  =  x
7 sneq 3651 . . . 4  |-  ( A  =  x  ->  { A }  =  { x } )
87eximi 1563 . . 3  |-  ( E. x  A  =  x  ->  E. x { A }  =  { x } )
96, 8ax-mp 8 . 2  |-  E. x { A }  =  {
x }
10 elsingles 23868 . 2  |-  ( { A }  e.  Singletons  <->  E. x { A }  =  { x } )
119, 10mpbir 200 1  |-  { A }  e.  Singletons
Colors of variables: wff set class
Syntax hints:   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640   Singletonscsingles 23793
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-eprel 4305  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fo 5261  df-fv 5263  df-1st 6122  df-2nd 6123  df-symdif 23773  df-txp 23806  df-singleton 23814  df-singles 23815
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