Theorem prcnel 3468

Description: A proper class doesn't belong to any class. (Contributed by Glauco Siliprandi, 17-Aug-2020.) (Proof shortened by AV, 14-Nov-2020.)

Assertion

Ref	Expression
prcnel	⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ 𝑉)

Proof of Theorem prcnel

Step	Hyp	Ref	Expression
1		elex 3463	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2	1	con3i 154	1 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ 𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2114  Vcvv 3442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444

This theorem is referenced by:  suppco  8160  fundmge2nop0  14439  fun2dmnop0  14441  vtxval  29091  iedgval  29092  xoromon  35272  fmlafvel  35607  isinf2  37687  eliin2f  45492  dfatprc  47519  afvprc  47533  afv2prc  47615