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Theorem nbgr0vtx 29116
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 4507 . . 3 𝑛 ∈ ∅ ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒
2 difeq1 4110 . . . . 5 ((Vtx‘𝐺) = ∅ → ((Vtx‘𝐺) ∖ {𝐾}) = (∅ ∖ {𝐾}))
3 0dif 4396 . . . . 5 (∅ ∖ {𝐾}) = ∅
42, 3eqtrdi 2782 . . . 4 ((Vtx‘𝐺) = ∅ → ((Vtx‘𝐺) ∖ {𝐾}) = ∅)
54raleqdv 3319 . . 3 ((Vtx‘𝐺) = ∅ → (∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝐾}) ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒 ↔ ∀𝑛 ∈ ∅ ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒))
61, 5mpbiri 258 . 2 ((Vtx‘𝐺) = ∅ → ∀𝑛 ∈ ((Vtx‘𝐺) ∖ {𝐾}) ¬ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑛} ⊆ 𝑒)
76nbgr0vtxlem 29115 1 ((Vtx‘𝐺) = ∅ → (𝐺 NeighbVtx 𝐾) = ∅)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1533  wral 3055  wrex 3064  cdif 3940  wss 3943  c0 4317  {csn 4623  {cpr 4625  cfv 6536  (class class class)co 7404  Vtxcvtx 28759  Edgcedg 28810   NeighbVtx cnbgr 29092
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-nbgr 29093
This theorem is referenced by: (None)
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