Theorem nvel 5281

Description: The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.) Prove it without using vprc 5282, 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 5279	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
2		elisset 2851	. 2 ⊢ (V ∈ 𝐴 → ∃𝑥 𝑥 = V)
3	1, 2	mto 200	1 ⊢ ¬ V ∈ 𝐴

Syntax hints:  ¬ wn 3   = wceq 1567  ∃wex 1806   ∈ wcel 2149  Vcvv 3463

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 5258

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465

This theorem is referenced by:  vprc  5282  onvf1odlem1  35482  curryset  37466  currysetlem3  37469  eliuniincex  45714  eliincex  45715  nvelim  47744