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

Theorem snelsingles 25759
Description: A singleton is a member of the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
Hypothesis
Ref Expression
snelsingles.1  |-  A  e. 
_V
Assertion
Ref Expression
snelsingles  |-  { A }  e.  Singletons

Proof of Theorem snelsingles
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 snelsingles.1 . . . 4  |-  A  e. 
_V
2 isset 2952 . . . . 5  |-  ( A  e.  _V  <->  E. x  x  =  A )
3 eqcom 2437 . . . . . 6  |-  ( x  =  A  <->  A  =  x )
43exbii 1592 . . . . 5  |-  ( E. x  x  =  A  <->  E. x  A  =  x )
52, 4bitri 241 . . . 4  |-  ( A  e.  _V  <->  E. x  A  =  x )
61, 5mpbi 200 . . 3  |-  E. x  A  =  x
7 sneq 3817 . . 3  |-  ( A  =  x  ->  { A }  =  { x } )
86, 7eximii 1587 . 2  |-  E. x { A }  =  {
x }
9 elsingles 25755 . 2  |-  ( { A }  e.  Singletons  <->  E. x { A }  =  { x } )
108, 9mpbir 201 1  |-  { A }  e.  Singletons
Colors of variables: wff set class
Syntax hints:   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2948   {csn 3806   Singletonscsingles 25675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-eprel 4486  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-fo 5452  df-fv 5454  df-1st 6341  df-2nd 6342  df-symdif 25655  df-txp 25690  df-singleton 25698  df-singles 25699
  Copyright terms: Public domain W3C validator