Theorem prcnel 3434

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 3428	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2	1	con3i 152	1 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ 𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2166  Vcvv 3413

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-12 2222  ax-ext 2802

This theorem depends on definitions:  df-bi 199  df-an 387  df-tru 1662  df-ex 1881  df-sb 2070  df-clab 2811  df-cleq 2817  df-clel 2820  df-v 3415

This theorem is referenced by:  fundmge2nop0  13562  fun2dmnop0  13564  vtxval  26297  iedgval  26298  eliin2f  40101  dfatprc  42031  afvprc  42045  afv2prc  42127