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Theorem nbgr0vtx 29387
Description: In a null graph (with no vertices), all neighborhoods are empty. (Contributed by AV, 15-Nov-2020.) (Proof shortened by AV, 10-May-2025.)
Assertion
Ref Expression
nbgr0vtx ((Vtx‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅)

Proof of Theorem nbgr0vtx
StepHypRef Expression
1 nel02 4345 . . 3 ((Vtx‘𝐺) = ∅ → ¬ 𝐾 ∈ (Vtx‘𝐺))
2 df-nel 3045 . . 3 (𝐾 ∉ (Vtx‘𝐺) ↔ ¬ 𝐾 ∈ (Vtx‘𝐺))
31, 2sylibr 234 . 2 ((Vtx‘𝐺) = ∅ → 𝐾 ∉ (Vtx‘𝐺))
4 eqid 2735 . . 3 (Vtx‘𝐺) = (Vtx‘𝐺)
54nbgrnvtx0 29371 . 2 (𝐾 ∉ (Vtx‘𝐺) → (𝐺 NeighbVtx 𝐾) = ∅)
63, 5syl 17 1 ((Vtx‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wcel 2106  wnel 3044  c0 4339  cfv 6563  (class class class)co 7431  Vtxcvtx 29028   NeighbVtx cnbgr 29364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-nbgr 29365
This theorem is referenced by: (None)
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