Theorem vsn 48736

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 4716	. 2 ⊢ (¬ V ∈ V ↔ {V} = ∅)
3	1, 2	mpbi 230	1 ⊢ {V} = ∅

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2107  Vcvv 3479  ∅c0 4332  {csn 4625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-dif 3953  df-nul 4333  df-sn 4626

This theorem is referenced by:  mo0  48738  setc1oterm  49162