ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  snprc Structured version   GIF version

Theorem snprc 3394
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 3350 . . . 4 (x {A} ↔ x = A)
21exbii 1465 . . 3 (x x {A} ↔ x x = A)
32notbii 577 . 2 x x {A} ↔ ¬ x x = A)
4 eq0 3203 . . 3 ({A} = ∅ ↔ x ¬ x {A})
5 alnex 1357 . . 3 (x ¬ x {A} ↔ ¬ x x {A})
64, 5bitri 173 . 2 ({A} = ∅ ↔ ¬ x x {A})
7 isset 2526 . . 3 (A V ↔ x x = A)
87notbii 577 . 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 1217   = wceq 1219  wex 1350   wcel 1362  Vcvv 2522  c0 3188  {csn 3335
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 528  ax-in2 529  ax-io 613  ax-5 1305  ax-7 1306  ax-gen 1307  ax-ie1 1351  ax-ie2 1352  ax-8 1364  ax-10 1365  ax-11 1366  ax-i12 1367  ax-bnd 1368  ax-4 1369  ax-17 1388  ax-i9 1392  ax-ial 1396  ax-i5r 1397  ax-ext 1991
This theorem depends on definitions:  df-bi 110  df-tru 1222  df-fal 1225  df-nf 1319  df-sb 1615  df-clab 1996  df-cleq 2002  df-clel 2005  df-nfc 2136  df-v 2524  df-dif 2884  df-nul 3189  df-sn 3341
This theorem is referenced by:  prprc1  3437  prprc  3439  snexprc  3897  sucprc  4083
  Copyright terms: Public domain W3C validator