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Theorem nbgrnvtx0 40563
 Description: There are no neighbors of a class which is not a vertex. (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 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = ∅)

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 6143 . . . . . 6 𝐺 / 𝑔(Vtx‘𝑔) = (Vtx‘𝐺)
31, 2eqtr4i 2635 . . . . 5 𝑉 = 𝐺 / 𝑔(Vtx‘𝑔)
4 neleq2 2889 . . . . 5 (𝑉 = 𝐺 / 𝑔(Vtx‘𝑔) → (𝑁𝑉𝑁𝐺 / 𝑔(Vtx‘𝑔)))
53, 4ax-mp 5 . . . 4 (𝑁𝑉𝑁𝐺 / 𝑔(Vtx‘𝑔))
65biimpi 205 . . 3 (𝑁𝑉𝑁𝐺 / 𝑔(Vtx‘𝑔))
76olcd 407 . 2 (𝑁𝑉 → (𝐺 ∉ V ∨ 𝑁𝐺 / 𝑔(Vtx‘𝑔)))
8 df-nbgr 40554 . . 3 NeighbVtx = (𝑔 ∈ V, 𝑣 ∈ (Vtx‘𝑔) ↦ {𝑛 ∈ ((Vtx‘𝑔) ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝑔){𝑣, 𝑛} ⊆ 𝑒})
98mpt2xneldm 7225 . 2 ((𝐺 ∉ V ∨ 𝑁𝐺 / 𝑔(Vtx‘𝑔)) → (𝐺 NeighbVtx 𝑁) = ∅)
107, 9syl 17 1 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   = wceq 1475   ∉ wnel 2781  ∃wrex 2897  {crab 2900  Vcvv 3173  ⦋csb 3499   ∖ cdif 3537   ⊆ wss 3540  ∅c0 3874  {csn 4125  {cpr 4127  ‘cfv 5804  (class class class)co 6549  Vtxcvtx 25673  Edgcedga 25792   NeighbVtx cnbgr 40550 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-nbgr 40554 This theorem is referenced by:  nbuhgr  40565  nbumgr  40569  nbgr0vtxlem  40577  nbgr1vtx  40580
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