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Theorem snprc 3675
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 3627 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
21exbii 1616 . . 3  |-  ( E. x  x  e.  { A }  <->  E. x  x  =  A )
32notbii 669 . 2  |-  ( -. 
E. x  x  e. 
{ A }  <->  -.  E. x  x  =  A )
4 eq0 3456 . . 3  |-  ( { A }  =  (/)  <->  A. x  -.  x  e.  { A } )
5 alnex 1510 . . 3  |-  ( A. x  -.  x  e.  { A }  <->  -.  E. x  x  e.  { A } )
64, 5bitri 184 . 2  |-  ( { A }  =  (/)  <->  -.  E. x  x  e.  { A } )
7 isset 2758 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
87notbii 669 . 2  |-  ( -.  A  e.  _V  <->  -.  E. x  x  =  A )
93, 6, 83bitr4ri 213 1  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 105   A.wal 1362    = wceq 1364   E.wex 1503    e. wcel 2160   _Vcvv 2752   (/)c0 3437   {csn 3610
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-nul 3438  df-sn 3616
This theorem is referenced by:  prprc1  3718  prprc  3720  snexprc  4207  sucprc  4433  snnen2oprc  6892  unsnfidcex  6952
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