Theorem vsn 49475

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 5285	. 2 ⊢ ¬ V ∈ V
2		snprc 4688	. 2 ⊢ (¬ V ∈ V ↔ {V} = ∅)
3	1, 2	mpbi 233	1 ⊢ {V} = ∅

Syntax hints:  ¬ wn 3   = wceq 1567   ∈ wcel 2149  Vcvv 3463  ∅c0 4294  {csn 4594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5261

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-dif 3916  df-nul 4295  df-sn 4595

This theorem is referenced by:  mo0  49477  setc1oterm  50154