Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  snexprc GIF version

Theorem snexprc 4110
 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 3588 . . 3 𝐴 ∈ V ↔ {𝐴} = ∅)
21biimpi 119 . 2 𝐴 ∈ V → {𝐴} = ∅)
3 0ex 4055 . 2 ∅ ∈ V
42, 3eqeltrdi 2230 1 𝐴 ∈ V → {𝐴} ∈ V)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1331   ∈ wcel 1480  Vcvv 2686  ∅c0 3363  {csn 3527 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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-nul 4054 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-dif 3073  df-nul 3364  df-sn 3533 This theorem is referenced by:  notnotsnex  4111
 Copyright terms: Public domain W3C validator