Theorem nvel 5266

Description: The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.) Prove it without using vprc 5267, which is then proved as an instance of it. (Revised by BJ, 1-May-2026.)

Assertion

Ref	Expression
nvel	⊢ ¬ V ∈ 𝐴

Proof of Theorem nvel

Step	Hyp	Ref	Expression
1		vnex 5264	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
2		elisset 2843	. 2 ⊢ (V ∈ 𝐴 → ∃𝑥 𝑥 = V)
3	1, 2	mto 199	1 ⊢ ¬ V ∈ 𝐴

Syntax hints:  ¬ wn 3   = wceq 1559  ∃wex 1798   ∈ wcel 2141  Vcvv 3453

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5243

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455

This theorem is referenced by:  vprc  5267  onvf1odlem1  35407  curryset  37392  currysetlem3  37395  eliuniincex  45648  eliincex  45649  nvelim  47678