MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  snprc Unicode version

Theorem snprc 3669
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 3629 . . . 4  |-  ( x  e.  { A }  <->  x  =  A )
21exbii 1580 . . 3  |-  ( E. x  x  e.  { A }  <->  E. x  x  =  A )
3 neq0 3440 . . 3  |-  ( -. 
{ A }  =  (/)  <->  E. x  x  e.  { A } )
4 isset 2767 . . 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 2763   (/)c0 3430   {csn 3614
This theorem is referenced by:  prprc1  3710  prprc  3712  snexALT  4168  snex  4188  sucprc  4439  unisn2  4494  posn  4746  frsn  4748  relimasn  5024  elimasni  5028  dmsnsnsn  5138  fv2  5454  fvprc  5455  fconst5  5665  1stval  6058  2ndval  6059  ecexr  6633  snfi  6909  domunsn  6979  hashsnlei  11341  efgrelexlema  15020  unisnif  23839  funpartfv  23858  wopprc  26490  inisegn0  26507
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 2239
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 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-v 2765  df-dif 3130  df-nul 3431  df-sn 3620
  Copyright terms: Public domain W3C validator