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Theorem vsn 47496
Description: The singleton of the universal class is the empty set. (Contributed by Zhi Wang, 19-Sep-2024.)
Assertion
Ref Expression
vsn {V} = ∅

Proof of Theorem vsn
StepHypRef Expression
1 vprc 5316 . 2 ¬ V ∈ V
2 snprc 4722 . 2 (¬ V ∈ V ↔ {V} = ∅)
31, 2mpbi 229 1 {V} = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2107  Vcvv 3475  c0 4323  {csn 4629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3952  df-nul 4324  df-sn 4630
This theorem is referenced by:  mo0  47498
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