MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nbgrnvtx0 Structured version   Visualization version   GIF version

Theorem nbgrnvtx0 27048
Description: If a class 𝑋 is not a vertex of a graph 𝐺, then it has no neighbors in 𝐺. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.)
Hypothesis
Ref Expression
nbgrel.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
nbgrnvtx0 (𝑋𝑉 → (𝐺 NeighbVtx 𝑋) = ∅)

Proof of Theorem nbgrnvtx0
Dummy variables 𝑒 𝑔 𝑛 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nbgrel.v . . . . . 6 𝑉 = (Vtx‘𝐺)
2 csbfv 6708 . . . . . 6 𝐺 / 𝑔(Vtx‘𝑔) = (Vtx‘𝐺)
31, 2eqtr4i 2844 . . . . 5 𝑉 = 𝐺 / 𝑔(Vtx‘𝑔)
4 neleq2 3126 . . . . 5 (𝑉 = 𝐺 / 𝑔(Vtx‘𝑔) → (𝑋𝑉𝑋𝐺 / 𝑔(Vtx‘𝑔)))
53, 4ax-mp 5 . . . 4 (𝑋𝑉𝑋𝐺 / 𝑔(Vtx‘𝑔))
65biimpi 217 . . 3 (𝑋𝑉𝑋𝐺 / 𝑔(Vtx‘𝑔))
76olcd 870 . 2 (𝑋𝑉 → (𝐺 ∉ V ∨ 𝑋𝐺 / 𝑔(Vtx‘𝑔)))
8 df-nbgr 27042 . . 3 NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})
98mpoxneldm 7867 . 2 ((𝐺 ∉ V ∨ 𝑋𝐺 / 𝑔(Vtx‘𝑔)) → (𝐺 NeighbVtx 𝑋) = ∅)
107, 9syl 17 1 (𝑋𝑉 → (𝐺 NeighbVtx 𝑋) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wo 841   = wceq 1528  wnel 3120  wrex 3136  {crab 3139  Vcvv 3492  csb 3880  cdif 3930  wss 3933  c0 4288  {csn 4557  {cpr 4559  cfv 6348  (class class class)co 7145  Vtxcvtx 26708  Edgcedg 26759   NeighbVtx cnbgr 27041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-nel 3121  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7678  df-2nd 7679  df-nbgr 27042
This theorem is referenced by:  nbuhgr  27052  nbumgr  27056  nbgr0vtxlem  27064  nbgr1vtx  27067
  Copyright terms: Public domain W3C validator