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Theorem snprc 3435
 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

Proof of Theorem snprc
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 velsn 3392 . . . 4
21exbii 1496 . . 3
32notbii 594 . 2
4 eq0 3239 . . 3
5 alnex 1388 . . 3
64, 5bitri 173 . 2
7 isset 2561 . . 3
87notbii 594 . 2
93, 6, 83bitr4ri 202 1
 Colors of variables: wff set class Syntax hints:   wn 3   wb 98  wal 1241   wceq 1243  wex 1381   wcel 1393  cvv 2557  c0 3224  csn 3375 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-dif 2920  df-nul 3225  df-sn 3381 This theorem is referenced by:  prprc1  3478  prprc  3480  snexprc  3938  sucprc  4149  snnen2oprc  6323
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