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Theorem vtxvalprc 26836
 Description: Degenerated case 4 for vertices: The set of vertices of a proper class is the empty set. (Contributed by AV, 12-Oct-2020.)
Assertion
Ref Expression
vtxvalprc (𝐶 ∉ V → (Vtx‘𝐶) = ∅)

Proof of Theorem vtxvalprc
StepHypRef Expression
1 df-nel 3116 . 2 (𝐶 ∉ V ↔ ¬ 𝐶 ∈ V)
2 fvprc 6645 . 2 𝐶 ∈ V → (Vtx‘𝐶) = ∅)
31, 2sylbi 220 1 (𝐶 ∉ V → (Vtx‘𝐶) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   = wceq 1538   ∈ wcel 2114   ∉ wnel 3115  Vcvv 3469  ∅c0 4265  ‘cfv 6334  Vtxcvtx 26787 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-nul 5186  ax-pow 5243 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-nel 3116  df-ral 3135  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-br 5043  df-iota 6293  df-fv 6342 This theorem is referenced by:  wlk0prc  27441
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