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Theorem snprc 3481
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 3439 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
21exbii 1537 . . 3  |-  ( E. x  x  e.  { A }  <->  E. x  x  =  A )
32notbii 627 . 2  |-  ( -. 
E. x  x  e. 
{ A }  <->  -.  E. x  x  =  A )
4 eq0 3284 . . 3  |-  ( { A }  =  (/)  <->  A. x  -.  x  e.  { A } )
5 alnex 1429 . . 3  |-  ( A. x  -.  x  e.  { A }  <->  -.  E. x  x  e.  { A } )
64, 5bitri 182 . 2  |-  ( { A }  =  (/)  <->  -.  E. x  x  e.  { A } )
7 isset 2616 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
87notbii 627 . 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 1283    = wceq 1285   E.wex 1422    e. wcel 1434   _Vcvv 2612   (/)c0 3269   {csn 3422
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-dif 2986  df-nul 3270  df-sn 3428
This theorem is referenced by:  prprc1  3524  prprc  3526  snexprc  3985  sucprc  4203  snnen2oprc  6506  unsnfidcex  6557
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