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Theorem snprc 3426
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 3382 . . . 4 (x {A} ↔ x = A)
21exbii 1493 . . 3 (x x {A} ↔ x x = A)
32notbii 593 . 2 x x {A} ↔ ¬ x x = A)
4 eq0 3233 . . 3 ({A} = ∅ ↔ x ¬ x {A})
5 alnex 1385 . . 3 (x ¬ x {A} ↔ ¬ x x {A})
64, 5bitri 173 . 2 ({A} = ∅ ↔ ¬ x x {A})
7 isset 2555 . . 3 (A V ↔ x x = A)
87notbii 593 . 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 1240   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  c0 3218  {csn 3367
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 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-nul 3219  df-sn 3373
This theorem is referenced by:  prprc1  3469  prprc  3471  snexprc  3929  sucprc  4115
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