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Theorem clnbgrnvtx0 47819
Description: If a class 𝑋 is not a vertex of a graph 𝐺, then it has an empty closed neighborhood in 𝐺. (Contributed by AV, 8-May-2025.)
Hypothesis
Ref Expression
clnbgrel.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
clnbgrnvtx0 (𝑋𝑉 → (𝐺 ClNeighbVtx 𝑋) = ∅)

Proof of Theorem clnbgrnvtx0
Dummy variables 𝑒 𝑔 𝑛 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 clnbgrel.v . . . . . 6 𝑉 = (Vtx‘𝐺)
2 csbfv 6955 . . . . . 6 𝐺 / 𝑔(Vtx‘𝑔) = (Vtx‘𝐺)
31, 2eqtr4i 2767 . . . . 5 𝑉 = 𝐺 / 𝑔(Vtx‘𝑔)
4 neleq2 3052 . . . . 5 (𝑉 = 𝐺 / 𝑔(Vtx‘𝑔) → (𝑋𝑉𝑋𝐺 / 𝑔(Vtx‘𝑔)))
53, 4ax-mp 5 . . . 4 (𝑋𝑉𝑋𝐺 / 𝑔(Vtx‘𝑔))
65biimpi 216 . . 3 (𝑋𝑉𝑋𝐺 / 𝑔(Vtx‘𝑔))
76olcd 874 . 2 (𝑋𝑉 → (𝐺 ∉ V ∨ 𝑋𝐺 / 𝑔(Vtx‘𝑔)))
8 df-clnbgr 47811 . . 3 ClNeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ ({𝑣} ∪ {𝑛 ∈ (Vtx‘𝑔) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒}))
98mpoxneldm 8238 . 2 ((𝐺 ∉ V ∨ 𝑋𝐺 / 𝑔(Vtx‘𝑔)) → (𝐺 ClNeighbVtx 𝑋) = ∅)
107, 9syl 17 1 (𝑋𝑉 → (𝐺 ClNeighbVtx 𝑋) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wo 847   = wceq 1539  wnel 3045  wrex 3069  {crab 3435  Vcvv 3479  csb 3898  cun 3948  wss 3950  c0 4332  {csn 4625  {cpr 4627  cfv 6560  (class class class)co 7432  Vtxcvtx 29014  Edgcedg 29065   ClNeighbVtx cclnbgr 47810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-1st 8015  df-2nd 8016  df-clnbgr 47811
This theorem is referenced by:  clnbgr0vtx  47827
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