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Theorem snprc 3502
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 velsn 3458 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
21exbii 1541 . . 3  |-  ( E. x  x  e.  { A }  <->  E. x  x  =  A )
32notbii 629 . 2  |-  ( -. 
E. x  x  e. 
{ A }  <->  -.  E. x  x  =  A )
4 eq0 3299 . . 3  |-  ( { A }  =  (/)  <->  A. x  -.  x  e.  { A } )
5 alnex 1433 . . 3  |-  ( A. x  -.  x  e.  { A }  <->  -.  E. x  x  e.  { A } )
64, 5bitri 182 . 2  |-  ( { A }  =  (/)  <->  -.  E. x  x  e.  { A } )
7 isset 2625 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
87notbii 629 . 2  |-  ( -.  A  e.  _V  <->  -.  E. x  x  =  A )
93, 6, 83bitr4ri 211 1  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 103   A.wal 1287    = wceq 1289   E.wex 1426    e. wcel 1438   _Vcvv 2619   (/)c0 3284   {csn 3441
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 2999  df-nul 3285  df-sn 3447
This theorem is referenced by:  prprc1  3545  prprc  3547  snexprc  4012  sucprc  4230  snnen2oprc  6556  unsnfidcex  6610
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