Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dfsingles2 Unicode version

Theorem dfsingles2 24460
Description: Alternate definition of the class of all singletons. (Contributed by Scott Fenton, 20-Nov-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
dfsingles2  |-  Singletons  =  {
x  |  E. y  x  =  { y } }
Distinct variable group:    x, y

Proof of Theorem dfsingles2
StepHypRef Expression
1 elsingles 24457 . 2  |-  ( x  e.  Singletons 
<->  E. y  x  =  { y } )
21abbi2i 2394 1  |-  Singletons  =  {
x  |  E. y  x  =  { y } }
Colors of variables: wff set class
Syntax hints:   E.wex 1528    = wceq 1623   {cab 2269   {csn 3640   Singletonscsingles 24382
This theorem is referenced by:  dfiota3  24462
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
  Copyright terms: Public domain W3C validator