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Theorem prcnel 3191
 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 → ¬ 𝐴𝑉)

Proof of Theorem prcnel
StepHypRef Expression
1 elex 3185 . 2 (𝐴𝑉𝐴 ∈ V)
21con3i 149 1 𝐴 ∈ V → ¬ 𝐴𝑉)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 1977  Vcvv 3173 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-an 385  df-tru 1478  df-ex 1696  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175 This theorem is referenced by:  nelsnOLD  4160  fundmge2nop0  13129  fun2dmnop0  13131  eliin2f  38316
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