Theorem vtxvalprc 29132

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

Step	Hyp	Ref	Expression
1		df-nel 3039	. 2 ⊢ (𝐶 ∉ V ↔ ¬ 𝐶 ∈ V)
2		fvprc 6819	. 2 ⊢ (¬ 𝐶 ∈ V → (Vtx‘𝐶) = ∅)
3	1, 2	sylbi 218	1 ⊢ (𝐶 ∉ V → (Vtx‘𝐶) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1547   ∈ wcel 2119   ∉ wnel 3038  Vcvv 3431  ∅c0 4261  ‘cfv 6485  Vtxcvtx 29083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711  ax-nul 5228  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-ne 2935  df-nel 3039  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-iota 6441  df-fv 6493

This theorem is referenced by:  wlk0prc  29739