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Theorem prcnel 3517
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 3511 . 2 (𝐴𝑉𝐴 ∈ V)
21con3i 157 1 𝐴 ∈ V → ¬ 𝐴𝑉)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2107  Vcvv 3493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1774  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-v 3495
This theorem is referenced by:  suppco  7862  fundmge2nop0  13842  fun2dmnop0  13844  vtxval  26777  iedgval  26778  fmlafvel  32620  isinf2  34673  eliin2f  41355  dfatprc  43314  afvprc  43328  afv2prc  43410
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