Theorem nvel 5284

Description: The universal class does not belong to any class. (Contributed by FL, 31-Dec-2006.) Prove it without using vprc 5285, 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 5282	. 2 ⊢ ¬ ∃𝑥 𝑥 = V
2		elisset 2847	. 2 ⊢ (V ∈ 𝐴 → ∃𝑥 𝑥 = V)
3	1, 2	mto 200	1 ⊢ ¬ V ∈ 𝐴

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570  ∃wex 1812   ∈ wcel 2146  Vcvv 3457

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459

This theorem is used by:  vprc  5285  onvf1odlem1  35644  curryset  37639  currysetlem3  37642  eliuniincex  45885  eliincex  45886  nvelim  47918