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Theorem prcnel 3505
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 3499 . 2 (𝐴𝑉𝐴 ∈ V)
21con3i 154 1 𝐴 ∈ V → ¬ 𝐴𝑉)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wcel 2106  Vcvv 3478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480
This theorem is referenced by:  suppco  8230  fundmge2nop0  14538  fun2dmnop0  14540  vtxval  29032  iedgval  29033  fmlafvel  35370  isinf2  37388  eliin2f  45044  dfatprc  47080  afvprc  47094  afv2prc  47176
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