Theorem vtxvalprc 29130

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 3038	. 2 ⊢ (𝐶 ∉ V ↔ ¬ 𝐶 ∈ V)
2		fvprc 6834	. 2 ⊢ (¬ 𝐶 ∈ V → (Vtx‘𝐶) = ∅)
3	1, 2	sylbi 217	1 ⊢ (𝐶 ∉ V → (Vtx‘𝐶) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2114   ∉ wnel 3037  Vcvv 3442  ∅c0 4287  ‘cfv 6500  Vtxcvtx 29081

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-nul 5253  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-nel 3038  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508

This theorem is referenced by:  wlk0prc  29738