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Theorem vtxvalprc 26157
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 3046 . 2 (𝐶 ∉ V ↔ ¬ 𝐶 ∈ V)
2 fvprc 6326 . 2 𝐶 ∈ V → (Vtx‘𝐶) = ∅)
31, 2sylbi 207 1 (𝐶 ∉ V → (Vtx‘𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1630  wcel 2144  wnel 3045  Vcvv 3349  c0 4061  cfv 6031  Vtxcvtx 26094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-nul 4920  ax-pow 4971
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-nel 3046  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-iota 5994  df-fv 6039
This theorem is referenced by:  wlk0prc  26784
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