Theorem prcnel 3469

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 3465	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2	1	con3i 154	1 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ 𝑉)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2132  Vcvv 3444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-ext 2724

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1553  df-ex 1790  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-v 3446

This theorem is referenced by:  suppco  8170  fundmge2nop0  14501  fun2dmnop0  14503  vtxval  29136  iedgval  29137  xoromon  35330  fmlafvel  35673  isinf2  37837  eliin2f  45620  dfatprc  47662  afvprc  47676  afv2prc  47758