MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uvtx0 Structured version   Visualization version   GIF version

Theorem uvtx0 27509
Description: There is no universal vertex if there is no vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 30-Oct-2020.) (Proof shortened by AV, 14-Feb-2022.)
Hypothesis
Ref Expression
uvtxel.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
uvtx0 (𝑉 = ∅ → (UnivVtx‘𝐺) = ∅)

Proof of Theorem uvtx0
Dummy variables 𝑛 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uvtxel.v . . 3 𝑉 = (Vtx‘𝐺)
21uvtxval 27502 . 2 (UnivVtx‘𝐺) = {𝑣𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)}
3 rabeq 3407 . . 3 (𝑉 = ∅ → {𝑣𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} = {𝑣 ∈ ∅ ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)})
4 rab0 4312 . . 3 {𝑣 ∈ ∅ ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} = ∅
53, 4eqtrdi 2795 . 2 (𝑉 = ∅ → {𝑣𝑉 ∣ ∀𝑛 ∈ (𝑉 ∖ {𝑣})𝑛 ∈ (𝐺 NeighbVtx 𝑣)} = ∅)
62, 5syl5eq 2791 1 (𝑉 = ∅ → (UnivVtx‘𝐺) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1543  wcel 2111  wral 3062  {crab 3066  cdif 3878  c0 4252  {csn 4556  cfv 6398  (class class class)co 7232  Vtxcvtx 27114   NeighbVtx cnbgr 27447  UnivVtxcuvtx 27500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5207  ax-nul 5214  ax-pr 5337
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3423  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4835  df-br 5069  df-opab 5131  df-mpt 5151  df-id 5470  df-xp 5572  df-rel 5573  df-cnv 5574  df-co 5575  df-dm 5576  df-iota 6356  df-fun 6400  df-fv 6406  df-ov 7235  df-uvtx 27501
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator