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Theorem elsingles 29543
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 3104 . 2  |-  ( A  e.  Singletons  ->  A  e.  _V )
2 snex 4678 . . . 4  |-  { x }  e.  _V
3 eleq1 2515 . . . 4  |-  ( A  =  { x }  ->  ( A  e.  _V  <->  { x }  e.  _V ) )
42, 3mpbiri 233 . . 3  |-  ( A  =  { x }  ->  A  e.  _V )
54exlimiv 1709 . 2  |-  ( E. x  A  =  {
x }  ->  A  e.  _V )
6 eleq1 2515 . . 3  |-  ( y  =  A  ->  (
y  e.  Singletons  <->  A  e.  Singletons ) )
7 eqeq1 2447 . . . 4  |-  ( y  =  A  ->  (
y  =  { x } 
<->  A  =  { x } ) )
87exbidv 1701 . . 3  |-  ( y  =  A  ->  ( E. x  y  =  { x }  <->  E. x  A  =  { x } ) )
9 df-singles 29487 . . . . 5  |-  Singletons  =  ran Singleton
109eleq2i 2521 . . . 4  |-  ( y  e.  Singletons 
<->  y  e.  ran Singleton )
11 vex 3098 . . . . 5  |-  y  e. 
_V
1211elrn 5233 . . . 4  |-  ( y  e.  ran Singleton  <->  E. x  xSingleton y
)
13 vex 3098 . . . . . 6  |-  x  e. 
_V
1413, 11brsingle 29542 . . . . 5  |-  ( xSingleton
y  <->  y  =  {
x } )
1514exbii 1654 . . . 4  |-  ( E. x  xSingleton y  <->  E. x  y  =  { x } )
1610, 12, 153bitri 271 . . 3  |-  ( y  e.  Singletons 
<->  E. x  y  =  { x } )
176, 8, 16vtoclbg 3154 . 2  |-  ( A  e.  _V  ->  ( A  e.  Singletons 
<->  E. x  A  =  { x } ) )
181, 5, 17pm5.21nii 353 1  |-  ( A  e.  Singletons 
<->  E. x  A  =  { x } )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1383   E.wex 1599    e. wcel 1804   _Vcvv 3095   {csn 4014   class class class wbr 4437   ran crn 4990  Singletoncsingle 29462   Singletonscsingles 29463
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-eprel 4781  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-fo 5584  df-fv 5586  df-1st 6785  df-2nd 6786  df-symdif 29443  df-txp 29478  df-singleton 29486  df-singles 29487
This theorem is referenced by:  dfsingles2  29546  snelsingles  29547  funpartlem  29567
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