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Theorem snexprc 3938
 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 3435 . . 3 𝐴 ∈ V ↔ {𝐴} = ∅)
21biimpi 113 . 2 𝐴 ∈ V → {𝐴} = ∅)
3 0ex 3884 . 2 ∅ ∈ V
42, 3syl6eqel 2128 1 𝐴 ∈ V → {𝐴} ∈ V)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1243   ∈ wcel 1393  Vcvv 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  ax-nul 3883 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: (None)
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