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Theorem vtxvalprc 29063
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 6897 . 2 𝐶 ∈ V → (Vtx‘𝐶) = ∅)
31, 2sylbi 217 1 (𝐶 ∉ V → (Vtx‘𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1539  wcel 2107  wnel 3045  Vcvv 3479  c0 4332  cfv 6560  Vtxcvtx 29014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568
This theorem is referenced by:  wlk0prc  29673
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