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Theorem snprc 3695
Description: The singleton of a proper class (one that doesn't exist) is the empty set. Theorem 7.2 of [Quine] p. 48. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
snprc  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )

Proof of Theorem snprc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 elsn 3655 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
21exbii 1569 . . 3  |-  ( E. x  x  e.  { A }  <->  E. x  x  =  A )
3 neq0 3465 . . 3  |-  ( -. 
{ A }  =  (/)  <->  E. x  x  e.  { A } )
4 isset 2792 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
52, 3, 43bitr4i 268 . 2  |-  ( -. 
{ A }  =  (/)  <->  A  e.  _V )
65con1bii 321 1  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788   (/)c0 3455   {csn 3640
This theorem is referenced by:  prprc1  3736  prprc  3738  snexALT  4196  snex  4216  sucprc  4467  unisn2  4522  posn  4758  frsn  4760  relimasn  5036  elimasni  5040  dmsnsnsn  5151  dffv3  5521  fconst5  5731  1stval  6124  2ndval  6125  ecexr  6665  snfi  6941  domunsn  7011  hashsnlei  11376  efgrelexlema  15058  eldm3  24119  unisnif  24464  funpartfv  24483  wopprc  27123  inisegn0  27140  elprchashprn2  28088  usgra1v  28126  cusgra1v  28157
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-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-dif 3155  df-nul 3456  df-sn 3646
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