Theorem vsn 46157

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 5239	. 2 ⊢ ¬ V ∈ V
2		snprc 4653	. 2 ⊢ (¬ V ∈ V ↔ {V} = ∅)
3	1, 2	mpbi 229	1 ⊢ {V} = ∅

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2106  Vcvv 3432  ∅c0 4256  {csn 4561

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-dif 3890  df-nul 4257  df-sn 4562

This theorem is referenced by:  mo0  46159