Theorem nvel 5241

Description: The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.) Prove it without using vprc 5242, 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 5239	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
2		elisset 2821	. 2 ⊢ (V ∈ 𝐴 → ∃𝑥 𝑥 = V)
3	1, 2	mto 198	1 ⊢ ¬ V ∈ 𝐴

Syntax hints:  ¬ wn 3   = wceq 1547  ∃wex 1786   ∈ wcel 2119  Vcvv 3431

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 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-sep 5218

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433

This theorem is referenced by:  vprc  5242  onvf1odlem1  35331  curryset  37299  currysetlem3  37302  eliuniincex  45556  eliincex  45557  nvelim  47586