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Theorem elsingles 24528
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 2809 . 2  |-  ( A  e.  Singletons  ->  A  e.  _V )
2 snex 4232 . . . 4  |-  { x }  e.  _V
3 eleq1 2356 . . . 4  |-  ( A  =  { x }  ->  ( A  e.  _V  <->  { x }  e.  _V ) )
42, 3mpbiri 224 . . 3  |-  ( A  =  { x }  ->  A  e.  _V )
54exlimiv 1624 . 2  |-  ( E. x  A  =  {
x }  ->  A  e.  _V )
6 eleq1 2356 . . 3  |-  ( y  =  A  ->  (
y  e.  Singletons  <->  A  e.  Singletons ) )
7 eqeq1 2302 . . . 4  |-  ( y  =  A  ->  (
y  =  { x } 
<->  A  =  { x } ) )
87exbidv 1616 . . 3  |-  ( y  =  A  ->  ( E. x  y  =  { x }  <->  E. x  A  =  { x } ) )
9 df-singles 24475 . . . . 5  |-  Singletons  =  ran Singleton
109eleq2i 2360 . . . 4  |-  ( y  e.  Singletons 
<->  y  e.  ran Singleton )
11 vex 2804 . . . . 5  |-  y  e. 
_V
1211elrn 4935 . . . 4  |-  ( y  e.  ran Singleton  <->  E. x  xSingleton y
)
13 vex 2804 . . . . . 6  |-  x  e. 
_V
1413, 11brsingle 24527 . . . . 5  |-  ( xSingleton
y  <->  y  =  {
x } )
1514exbii 1572 . . . 4  |-  ( E. x  xSingleton y  <->  E. x  y  =  { x } )
1610, 12, 153bitri 262 . . 3  |-  ( y  e.  Singletons 
<->  E. x  y  =  { x } )
176, 8, 16vtoclbg 2857 . 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 1531    = wceq 1632    e. wcel 1696   _Vcvv 2801   {csn 3653   class class class wbr 4039   ran crn 4706  Singletoncsingle 24452   Singletonscsingles 24453
This theorem is referenced by:  dfsingles2  24531  snelsingles  24532  funpartlem  24552
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-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-eprel 4321  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-1st 6138  df-2nd 6139  df-symdif 24433  df-txp 24466  df-singleton 24474  df-singles 24475
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