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

Theorem nbgrnvtx0 29275
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 6951 . . . . . 6 𝐺 / 𝑔(Vtx‘𝑔) = (Vtx‘𝐺)
31, 2eqtr4i 2757 . . . . 5 𝑉 = 𝐺 / 𝑔(Vtx‘𝑔)
4 neleq2 3043 . . . . 5 (𝑉 = 𝐺 / 𝑔(Vtx‘𝑔) → (𝑋𝑉𝑋𝐺 / 𝑔(Vtx‘𝑔)))
53, 4ax-mp 5 . . . 4 (𝑋𝑉𝑋𝐺 / 𝑔(Vtx‘𝑔))
65biimpi 215 . . 3 (𝑋𝑉𝑋𝐺 / 𝑔(Vtx‘𝑔))
76olcd 872 . 2 (𝑋𝑉 → (𝐺 ∉ V ∨ 𝑋𝐺 / 𝑔(Vtx‘𝑔)))
8 df-nbgr 29269 . . 3 NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})
98mpoxneldm 8227 . 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 205  wo 845   = wceq 1534  wnel 3036  wrex 3060  {crab 3419  Vcvv 3462  csb 3892  cdif 3944  wss 3947  c0 4325  {csn 4633  {cpr 4635  cfv 6554  (class class class)co 7424  Vtxcvtx 28932  Edgcedg 28983   NeighbVtx cnbgr 29268
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-nel 3037  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-nbgr 29269
This theorem is referenced by:  nbuhgr  29279  nbumgr  29283  nbgr0vtx  29291  nbgr0edglem  29292  nbgr1vtx  29294
  Copyright terms: Public domain W3C validator