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Theorem nbgrnvtx0 27132
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 6694 . . . . . 6 𝐺 / 𝑔(Vtx‘𝑔) = (Vtx‘𝐺)
31, 2eqtr4i 2827 . . . . 5 𝑉 = 𝐺 / 𝑔(Vtx‘𝑔)
4 neleq2 3100 . . . . 5 (𝑉 = 𝐺 / 𝑔(Vtx‘𝑔) → (𝑋𝑉𝑋𝐺 / 𝑔(Vtx‘𝑔)))
53, 4ax-mp 5 . . . 4 (𝑋𝑉𝑋𝐺 / 𝑔(Vtx‘𝑔))
65biimpi 219 . . 3 (𝑋𝑉𝑋𝐺 / 𝑔(Vtx‘𝑔))
76olcd 871 . 2 (𝑋𝑉 → (𝐺 ∉ V ∨ 𝑋𝐺 / 𝑔(Vtx‘𝑔)))
8 df-nbgr 27126 . . 3 NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})
98mpoxneldm 7865 . 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 209  wo 844   = wceq 1538  wnel 3094  wrex 3110  {crab 3113  Vcvv 3444  csb 3831  cdif 3881  wss 3884  c0 4246  {csn 4528  {cpr 4530  cfv 6328  (class class class)co 7139  Vtxcvtx 26792  Edgcedg 26843   NeighbVtx cnbgr 27125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-nel 3095  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7142  df-oprab 7143  df-mpo 7144  df-1st 7675  df-2nd 7676  df-nbgr 27126
This theorem is referenced by:  nbuhgr  27136  nbumgr  27140  nbgr0vtxlem  27148  nbgr1vtx  27151
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