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Theorem nbgrcl 27169
 Description: If a class 𝑋 has at least one neighbor, this class must be a vertex. (Contributed by AV, 6-Jun-2021.) (Revised by AV, 12-Feb-2022.)
Hypothesis
Ref Expression
nbgrcl.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
nbgrcl (𝑁 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋𝑉)

Proof of Theorem nbgrcl
Dummy variables 𝑔 𝑒 𝑛 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nbgr 27167 . . 3 NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})
21mpoxeldm 7878 . 2 (𝑁 ∈ (𝐺 NeighbVtx 𝑋) → (𝐺 ∈ V ∧ 𝑋𝐺 / 𝑔(Vtx‘𝑔)))
3 csbfv 6700 . . . . 5 𝐺 / 𝑔(Vtx‘𝑔) = (Vtx‘𝐺)
4 nbgrcl.v . . . . 5 𝑉 = (Vtx‘𝐺)
53, 4eqtr4i 2824 . . . 4 𝐺 / 𝑔(Vtx‘𝑔) = 𝑉
65eleq2i 2881 . . 3 (𝑋𝐺 / 𝑔(Vtx‘𝑔) ↔ 𝑋𝑉)
76biimpi 219 . 2 (𝑋𝐺 / 𝑔(Vtx‘𝑔) → 𝑋𝑉)
82, 7simpl2im 507 1 (𝑁 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋𝑉)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  {crab 3110  Vcvv 3442  ⦋csb 3830   ∖ cdif 3880   ⊆ wss 3883  {csn 4528  {cpr 4530  ‘cfv 6332  (class class class)co 7145  Vtxcvtx 26833  Edgcedg 26884   NeighbVtx cnbgr 27166 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 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5171  ax-nul 5178  ax-pr 5299  ax-un 7454 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 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3444  df-sbc 3723  df-csb 3831  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4805  df-iun 4887  df-br 5035  df-opab 5097  df-mpt 5115  df-id 5429  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 6291  df-fun 6334  df-fv 6340  df-ov 7148  df-oprab 7149  df-mpo 7150  df-1st 7684  df-2nd 7685  df-nbgr 27167 This theorem is referenced by:  nbgrel  27174  frgrnbnb  28122
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