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Theorem snexprc 3929
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 3426 . . 3 A V ↔ {A} = ∅)
21biimpi 113 . 2 A V → {A} = ∅)
3 0ex 3875 . 2 V
42, 3syl6eqel 2125 1 A V → {A} V)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1242   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  ax-nul 3874
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: (None)
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