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Theorem nbgrnvtx0 26814
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 6539 . . . . . 6 𝐺 / 𝑔(Vtx‘𝑔) = (Vtx‘𝐺)
31, 2eqtr4i 2799 . . . . 5 𝑉 = 𝐺 / 𝑔(Vtx‘𝑔)
4 neleq2 3073 . . . . 5 (𝑉 = 𝐺 / 𝑔(Vtx‘𝑔) → (𝑋𝑉𝑋𝐺 / 𝑔(Vtx‘𝑔)))
53, 4ax-mp 5 . . . 4 (𝑋𝑉𝑋𝐺 / 𝑔(Vtx‘𝑔))
65biimpi 208 . . 3 (𝑋𝑉𝑋𝐺 / 𝑔(Vtx‘𝑔))
76olcd 860 . 2 (𝑋𝑉 → (𝐺 ∉ V ∨ 𝑋𝐺 / 𝑔(Vtx‘𝑔)))
8 df-nbgr 26808 . . 3 NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})
98mpoxneldm 7674 . 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 198  wo 833   = wceq 1507  wnel 3067  wrex 3083  {crab 3086  Vcvv 3409  csb 3782  cdif 3822  wss 3825  c0 4173  {csn 4435  {cpr 4437  cfv 6182  (class class class)co 6970  Vtxcvtx 26474  Edgcedg 26525   NeighbVtx cnbgr 26807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-fal 1520  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-nel 3068  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-csb 3783  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-iun 4788  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fv 6190  df-ov 6973  df-oprab 6974  df-mpo 6975  df-1st 7494  df-2nd 7495  df-nbgr 26808
This theorem is referenced by:  nbuhgr  26818  nbumgr  26822  nbgr0vtxlem  26830  nbgr1vtx  26833
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