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Theorem nbgr0vtx 26475
Description: In a null graph (with no vertices), all neighborhoods are empty. (Contributed by AV, 15-Nov-2020.)
Assertion
Ref Expression
nbgr0vtx ((Vtx‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅)

Proof of Theorem nbgr0vtx
Dummy variables 𝑒 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ral0 4217 . . 3 𝑛 ∈ ∅ ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒
2 difeq1 3872 . . . . 5 ((Vtx‘𝐺) = ∅ → ((Vtx‘𝐺) ∖ {𝐾}) = (∅ ∖ {𝐾}))
3 0dif 4121 . . . . 5 (∅ ∖ {𝐾}) = ∅
42, 3syl6eq 2821 . . . 4 ((Vtx‘𝐺) = ∅ → ((Vtx‘𝐺) ∖ {𝐾}) = ∅)
54raleqdv 3293 . . 3 ((Vtx‘𝐺) = ∅ → (∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝐾}) ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒 ↔ ∀𝑛 ∈ ∅ ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒))
61, 5mpbiri 248 . 2 ((Vtx‘𝐺) = ∅ → ∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝐾}) ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒)
76nbgr0vtxlem 26474 1 ((Vtx‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1631  wral 3061  wrex 3062  cdif 3720  wss 3723  c0 4063  {csn 4316  {cpr 4318  cfv 6031  (class class class)co 6793  Vtxcvtx 26095  Edgcedg 26160   NeighbVtx cnbgr 26447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fv 6039  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-1st 7315  df-2nd 7316  df-nbgr 26448
This theorem is referenced by: (None)
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