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Theorem snprc 3636
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 3596 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
21exbii 1580 . . 3  |-  ( E. x  x  e.  { A }  <->  E. x  x  =  A )
3 neq0 3407 . . 3  |-  ( -. 
{ A }  =  (/)  <->  E. x  x  e.  { A } )
4 isset 2744 . . 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 2740   (/)c0 3397   {csn 3581
This theorem is referenced by:  prprc1  3677  prprc  3679  snexALT  4134  snex  4154  sucprc  4404  unisn2  4459  posn  4711  frsn  4713  relimasn  4989  elimasni  4993  dmsnsnsn  5103  fv2  5419  fvprc  5420  fconst5  5630  1stval  6023  2ndval  6024  ecexr  6598  snfi  6874  domunsn  6944  hashsnlei  11306  efgrelexlema  14985  unisnif  23804  funpartfv  23823  wopprc  26455  inisegn0  26472
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 1927  ax-ext 2237
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 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-v 2742  df-dif 3097  df-nul 3398  df-sn 3587
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