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Theorem snexprc 3912
Description: A singleton whose element is a proper class is a set. The ¬ A V case of Theorem 7.12 of [Quine] p. 51, proved using only Extensionality, Power Set, and Separation. Replacement is not needed. (Contributed by Jim Kingdon, 1-Sep-2018.)
Assertion
Ref Expression
snexprc A V → {A} V)

Proof of Theorem snexprc
StepHypRef Expression
1 snprc 3409 . . 3 A V ↔ {A} = ∅)
21biimpi 113 . 2 A V → {A} = ∅)
3 0ex 3858 . 2 V
42, 3syl6eqel 2110 1 A V → {A} V)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1228   wcel 1374  Vcvv 2535  c0 3201  {csn 3350
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 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-nul 3857
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-fal 1234  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-dif 2897  df-nul 3202  df-sn 3356
This theorem is referenced by: (None)
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