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Theorem nbgranv0 25956
 Description: There are no neighbors of a class which is not a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.)
Assertion
Ref Expression
nbgranv0 (𝑁𝑉 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = ∅)

Proof of Theorem nbgranv0
Dummy variables 𝑔 𝑘 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-nbgra 25949 . 2 Neighbors = (𝑔 ∈ V, 𝑘 ∈ (1st𝑔) ↦ {𝑛 ∈ (1st𝑔) ∣ {𝑘, 𝑛} ∈ ran (2nd𝑔)})
21mpt2xopynvov0 7231 1 (𝑁𝑉 → (⟨𝑉, 𝐸⟩ Neighbors 𝑁) = ∅)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   ∉ wnel 2781  {crab 2900  ∅c0 3874  {cpr 4127  ⟨cop 4131  ran crn 5039  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058   Neighbors cnbgra 25946 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-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-nbgra 25949 This theorem is referenced by:  nbusgra  25957  nbgranself2  25965
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