Theorem vsn 47398

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

Step	Hyp	Ref	Expression
1		vprc 5314	. 2 ⊢ ¬ V ∈ V
2		snprc 4720	. 2 ⊢ (¬ V ∈ V ↔ {V} = ∅)
3	1, 2	mpbi 229	1 ⊢ {V} = ∅

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2107  Vcvv 3475  ∅c0 4321  {csn 4627

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 5298

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 3950  df-nul 4322  df-sn 4628

This theorem is referenced by:  mo0  47400