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Theorem elsingles 24457
Description: Membership in the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
Assertion
Ref Expression
elsingles  |-  ( A  e.  Singletons 
<->  E. x  A  =  { x } )
Distinct variable group:    x, A

Proof of Theorem elsingles
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 elex 2796 . 2  |-  ( A  e.  Singletons  ->  A  e.  _V )
2 snex 4216 . . . 4  |-  { x }  e.  _V
3 eleq1 2343 . . . 4  |-  ( A  =  { x }  ->  ( A  e.  _V  <->  { x }  e.  _V ) )
42, 3mpbiri 224 . . 3  |-  ( A  =  { x }  ->  A  e.  _V )
54exlimiv 1666 . 2  |-  ( E. x  A  =  {
x }  ->  A  e.  _V )
6 eleq1 2343 . . 3  |-  ( y  =  A  ->  (
y  e.  Singletons  <->  A  e.  Singletons ) )
7 eqeq1 2289 . . . 4  |-  ( y  =  A  ->  (
y  =  { x } 
<->  A  =  { x } ) )
87exbidv 1612 . . 3  |-  ( y  =  A  ->  ( E. x  y  =  { x }  <->  E. x  A  =  { x } ) )
9 df-singles 24404 . . . . 5  |-  Singletons  =  ran Singleton
109eleq2i 2347 . . . 4  |-  ( y  e.  Singletons 
<->  y  e.  ran Singleton )
11 vex 2791 . . . . 5  |-  y  e. 
_V
1211elrn 4919 . . . 4  |-  ( y  e.  ran Singleton  <->  E. x  xSingleton y
)
13 vex 2791 . . . . . 6  |-  x  e. 
_V
1413, 11brsingle 24456 . . . . 5  |-  ( xSingleton
y  <->  y  =  {
x } )
1514exbii 1569 . . . 4  |-  ( E. x  xSingleton y  <->  E. x  y  =  { x } )
1610, 12, 153bitri 262 . . 3  |-  ( y  e.  Singletons 
<->  E. x  y  =  { x } )
176, 8, 16vtoclbg 2844 . 2  |-  ( A  e.  _V  ->  ( A  e.  Singletons 
<->  E. x  A  =  { x } ) )
181, 5, 17pm5.21nii 342 1  |-  ( A  e.  Singletons 
<->  E. x  A  =  { x } )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788   {csn 3640   class class class wbr 4023   ran crn 4690  Singletoncsingle 24381   Singletonscsingles 24382
This theorem is referenced by:  dfsingles2  24460  snelsingles  24461  funpartfun  24481  funpartfv  24483
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 24362  df-txp 24395  df-singleton 24403  df-singles 24404
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