Theorem vsn 50062

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 5277	. 2 ⊢ ¬ V ∈ V
2		snprc 4677	. 2 ⊢ (¬ V ∈ V ↔ {V} = ∅)
3	1, 2	mpbi 233	1 ⊢ {V} = ∅

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1569   ∈ wcel 2146  Vcvv 3453  ∅c0 4280  {csn 4587

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3455  df-dif 4049  df-nul 4281  df-sn 4588

This theorem is used by:  mo0  50064  setc1oterm  50765