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Theorem snprc 3599
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
StepHypRef Expression
1 elsn 3559 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
21exbii 1580 . . 3  |-  ( E. x  x  e.  { A }  <->  E. x  x  =  A )
3 neq0 3372 . . 3  |-  ( -. 
{ A }  =  (/)  <->  E. x  x  e.  { A } )
4 isset 2731 . . 3  |-  ( A  e.  _V  <->  E. x  x  =  A )
52, 3, 43bitr4i 270 . 2  |-  ( -. 
{ A }  =  (/)  <->  A  e.  _V )
65con1bii 323 1  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    <-> wb 178   E.wex 1537    = wceq 1619    e. wcel 1621   _Vcvv 2727   (/)c0 3362   {csn 3544
This theorem is referenced by:  prprc1  3640  prprc  3642  snexALT  4090  snex  4110  sucprc  4360  unisn2  4413  posn  4665  frsn  4667  relimasn  4943  elimasni  4947  dmsnsnsn  5057  fv2  5373  fvprc  5374  fconst5  5583  1stval  5976  2ndval  5977  ecexr  6551  snfi  6826  domunsn  6896  hashsnlei  11253  efgrelexlema  14893  unisnif  23638  funpartfv  23657  wopprc  26289  inisegn0  26306
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-v 2729  df-dif 3081  df-nul 3363  df-sn 3550
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