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Theorem snprc 3595
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 3548 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
21exbii 1585 . . 3  |-  ( E. x  x  e.  { A }  <->  E. x  x  =  A )
32notbii 658 . 2  |-  ( -. 
E. x  x  e. 
{ A }  <->  -.  E. x  x  =  A )
4 eq0 3385 . . 3  |-  ( { A }  =  (/)  <->  A. x  -.  x  e.  { A } )
5 alnex 1476 . . 3  |-  ( A. x  -.  x  e.  { A }  <->  -.  E. x  x  e.  { A } )
64, 5bitri 183 . 2  |-  ( { A }  =  (/)  <->  -.  E. x  x  e.  { A } )
7 isset 2695 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
87notbii 658 . 2  |-  ( -.  A  e.  _V  <->  -.  E. x  x  =  A )
93, 6, 83bitr4ri 212 1  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 104   A.wal 1330    = wceq 1332   E.wex 1469    e. wcel 1481   _Vcvv 2689   (/)c0 3367   {csn 3531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-dif 3077  df-nul 3368  df-sn 3537
This theorem is referenced by:  prprc1  3638  prprc  3640  snexprc  4117  sucprc  4341  snnen2oprc  6761  unsnfidcex  6815
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