Theorem prcnel 3476

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 3472	. 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 2145  Vcvv 3451

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 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3453

This theorem is used by:  suppco  8223  fundmge2nop0  14647  fun2dmnop0  14649  vtxval  29578  iedgval  29579  ordprcon  35715  xoromon  35716  fmlafvel  36150  isinf2  38328  eliin2f  46118  dfatprc  48199  afvprc  48213  afv2prc  48295