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

Step	Hyp	Ref	Expression
1		vprc 5193	. 2 ⊢ ¬ V ∈ V
2		snprc 4618	. 2 ⊢ (¬ V ∈ V ↔ {V} = ∅)
3	1, 2	mpbi 233	1 ⊢ {V} = ∅

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