Theorem prcnel 3479

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 3475	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2	1	con3i 155	1 ⊢ (¬ 𝐴 ∈ V → ¬ 𝐴 ∈ 𝑉)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∈ wcel 2146  Vcvv 3454

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456

This theorem is used by:  suppco  8237  fundmge2nop0  14660  fun2dmnop0  14662  vtxval  29593  iedgval  29594  ordprcon  35776  xoromon  35777  fmlafvel  36274  isinf2  38474  eliin2f  46264  dfatprc  48345  afvprc  48359  afv2prc  48441