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Theorem snexprc 4204
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 3672 . . 3 𝐴 ∈ V ↔ {𝐴} = ∅)
21biimpi 120 . 2 𝐴 ∈ V → {𝐴} = ∅)
3 0ex 4145 . 2 ∅ ∈ V
42, 3eqeltrdi 2280 1 𝐴 ∈ V → {𝐴} ∈ V)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1364  wcel 2160  Vcvv 2752  c0 3437  {csn 3607
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-nul 4144
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-nul 3438  df-sn 3613
This theorem is referenced by:  notnotsnex  4205
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