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Theorem vsn 45736
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 5193 . 2 ¬ V ∈ V
2 snprc 4618 . 2 (¬ V ∈ V ↔ {V} = ∅)
31, 2mpbi 233 1 {V} = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2114  Vcvv 3400  c0 4221  {csn 4526
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2711  ax-sep 5177
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2718  df-cleq 2731  df-clel 2812  df-v 3402  df-dif 3856  df-nul 4222  df-sn 4527
This theorem is referenced by:  mo0  45738
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