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Theorem snprc 3939
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, 21-Jun-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 3891 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
21exbii 1634 . . 3  |-  ( E. x  x  e.  { A }  <->  E. x  x  =  A )
3 neq0 3647 . . 3  |-  ( -. 
{ A }  =  (/)  <->  E. x  x  e.  { A } )
4 isset 2976 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
52, 3, 43bitr4i 277 . 2  |-  ( -. 
{ A }  =  (/)  <->  A  e.  _V )
65con1bii 331 1  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    = wceq 1369   E.wex 1586    e. wcel 1756   _Vcvv 2972   (/)c0 3637   {csn 3877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-v 2974  df-dif 3331  df-nul 3638  df-sn 3878
This theorem is referenced by:  snnzb  3940  rabsnif  3944  prprc1  3985  prprc  3987  unisn2  4428  snexALT  4478  snex  4533  sucprc  4794  posn  4907  frsn  4909  relimasn  5192  elimasni  5196  dmsnsnsn  5317  dffv3  5687  fconst5  5935  1stval  6579  2ndval  6580  ecexr  7106  snfi  7390  domunsn  7461  hashrabrsn  12137  elprchashprn2  12156  hashsnlei  12170  hash2pwpr  12182  efgrelexlema  16246  usgra1v  23308  cusgra1v  23369  1conngra  23561  eldm3  27572  opelco3  27589  fvsingle  27951  unisnif  27956  funpartlem  27973  wopprc  29379  inisegn0  29396  hashrabsn01  30232  hashrabsn1  30233  snnen2o  30738  bj-sngltag  32476
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