Theorem vtxvalprc 29009

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 3030	. 2 ⊢ (𝐶 ∉ V ↔ ¬ 𝐶 ∈ V)
2		fvprc 6818	. 2 ⊢ (¬ 𝐶 ∈ V → (Vtx‘𝐶) = ∅)
3	1, 2	sylbi 217	1 ⊢ (𝐶 ∉ V → (Vtx‘𝐶) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109   ∉ wnel 3029  Vcvv 3438  ∅c0 4286  ‘cfv 6486  Vtxcvtx 28960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nel 3030  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-iota 6442  df-fv 6494

This theorem is referenced by:  wlk0prc  29617