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Theorem elsingles 25765
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 2966 . 2  |-  ( A  e.  Singletons  ->  A  e.  _V )
2 snex 4407 . . . 4  |-  { x }  e.  _V
3 eleq1 2498 . . . 4  |-  ( A  =  { x }  ->  ( A  e.  _V  <->  { x }  e.  _V ) )
42, 3mpbiri 226 . . 3  |-  ( A  =  { x }  ->  A  e.  _V )
54exlimiv 1645 . 2  |-  ( E. x  A  =  {
x }  ->  A  e.  _V )
6 eleq1 2498 . . 3  |-  ( y  =  A  ->  (
y  e.  Singletons  <->  A  e.  Singletons ) )
7 eqeq1 2444 . . . 4  |-  ( y  =  A  ->  (
y  =  { x } 
<->  A  =  { x } ) )
87exbidv 1637 . . 3  |-  ( y  =  A  ->  ( E. x  y  =  { x }  <->  E. x  A  =  { x } ) )
9 df-singles 25709 . . . . 5  |-  Singletons  =  ran Singleton
109eleq2i 2502 . . . 4  |-  ( y  e.  Singletons 
<->  y  e.  ran Singleton )
11 vex 2961 . . . . 5  |-  y  e. 
_V
1211elrn 5112 . . . 4  |-  ( y  e.  ran Singleton  <->  E. x  xSingleton y
)
13 vex 2961 . . . . . 6  |-  x  e. 
_V
1413, 11brsingle 25764 . . . . 5  |-  ( xSingleton
y  <->  y  =  {
x } )
1514exbii 1593 . . . 4  |-  ( E. x  xSingleton y  <->  E. x  y  =  { x } )
1610, 12, 153bitri 264 . . 3  |-  ( y  e.  Singletons 
<->  E. x  y  =  { x } )
176, 8, 16vtoclbg 3014 . 2  |-  ( A  e.  _V  ->  ( A  e.  Singletons 
<->  E. x  A  =  { x } ) )
181, 5, 17pm5.21nii 344 1  |-  ( A  e.  Singletons 
<->  E. x  A  =  { x } )
Colors of variables: wff set class
Syntax hints:    <-> wb 178   E.wex 1551    = wceq 1653    e. wcel 1726   _Vcvv 2958   {csn 3816   class class class wbr 4214   ran crn 4881  Singletoncsingle 25684   Singletonscsingles 25685
This theorem is referenced by:  dfsingles2  25768  snelsingles  25769  funpartlem  25789
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-mpt 4270  df-eprel 4496  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fo 5462  df-fv 5464  df-1st 6351  df-2nd 6352  df-symdif 25665  df-txp 25700  df-singleton 25708  df-singles 25709
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