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Theorem snprc 3387
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 V ↔ {A} = ∅)

Proof of Theorem snprc
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 elsn 3343 . . . 4 (x {A} ↔ x = A)
21exbii 1480 . . 3 (x x {A} ↔ x x = A)
32notbii 581 . 2 x x {A} ↔ ¬ x x = A)
4 eq0 3218 . . 3 ({A} = ∅ ↔ x ¬ x {A})
5 alnex 1369 . . 3 (x ¬ x {A} ↔ ¬ x x {A})
64, 5bitri 173 . 2 ({A} = ∅ ↔ ¬ x x {A})
7 isset 2538 . . 3 (A V ↔ x x = A)
87notbii 581 . 2 A V ↔ ¬ x x = A)
93, 6, 83bitr4ri 202 1 A V ↔ {A} = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wb 98  wal 1315  wex 1362   = wceq 1374   wcel 1376  Vcvv 2534  c0 3203  {csn 3328
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 532  ax-in2 533  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1378  ax-10 1379  ax-11 1380  ax-i12 1381  ax-bnd 1382  ax-4 1383  ax-17 1402  ax-i9 1406  ax-ial 1411  ax-i5r 1412  ax-ext 2005
This theorem depends on definitions:  df-bi 110  df-tru 1232  df-fal 1233  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-v 2536  df-dif 2899  df-nul 3204  df-sn 3334
This theorem is referenced by:  prprc1  3430  prprc  3432  snexprc  3891  sucprc  4074
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