MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  vtxvalprc Structured version   Visualization version   GIF version

Theorem vtxvalprc 27704
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 3047 . 2 (𝐶 ∉ V ↔ ¬ 𝐶 ∈ V)
2 fvprc 6817 . 2 (¬ 𝐶 ∈ V → (Vtx‘𝐶) = ∅)
31, 2sylbi 216 1 (𝐶 ∉ V → (Vtx‘𝐶) = ∅)
Colors of variables: wff setvar class
Syntax hints:  Â¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2105   ∉ wnel 3046  Vcvv 3441  âˆ…c0 4269  â€˜cfv 6479  Vtxcvtx 27655
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-nul 5250  ax-pr 5372
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3404  df-v 3443  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-br 5093  df-iota 6431  df-fv 6487
This theorem is referenced by:  wlk0prc  28310
  Copyright terms: Public domain W3C validator