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Theorem snexprc 4170
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 3646 . . 3 𝐴 ∈ V ↔ {𝐴} = ∅)
21biimpi 119 . 2 𝐴 ∈ V → {𝐴} = ∅)
3 0ex 4114 . 2 ∅ ∈ V
42, 3eqeltrdi 2261 1 𝐴 ∈ V → {𝐴} ∈ V)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1348  wcel 2141  Vcvv 2730  c0 3414  {csn 3581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-nul 4113
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-dif 3123  df-nul 3415  df-sn 3587
This theorem is referenced by:  notnotsnex  4171
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