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Theorem nbgrnself 28605
Description: A vertex in a graph is not a neighbor of itself. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 3-Nov-2020.) (Revised by AV, 21-Mar-2021.)
Hypothesis
Ref Expression
nbgrnself.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
nbgrnself 𝑣𝑉 𝑣 ∉ (𝐺 NeighbVtx 𝑣)
Distinct variable group:   𝑣,𝑉
Allowed substitution hint:   𝐺(𝑣)

Proof of Theorem nbgrnself
Dummy variables 𝑒 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 neldifsnd 4795 . . . . 5 (𝑣𝑉 → ¬ 𝑣 ∈ (𝑉 ∖ {𝑣}))
21intnanrd 490 . . . 4 (𝑣𝑉 → ¬ (𝑣 ∈ (𝑉 ∖ {𝑣}) ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
3 df-nel 3047 . . . . 5 (𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ ¬ 𝑣 ∈ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒})
4 preq2 4737 . . . . . . . 8 (𝑛 = 𝑣 → {𝑣, 𝑛} = {𝑣, 𝑣})
54sseq1d 4012 . . . . . . 7 (𝑛 = 𝑣 → ({𝑣, 𝑛} ⊆ 𝑒 ↔ {𝑣, 𝑣} ⊆ 𝑒))
65rexbidv 3178 . . . . . 6 (𝑛 = 𝑣 → (∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
76elrab 3682 . . . . 5 (𝑣 ∈ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ (𝑣 ∈ (𝑉 ∖ {𝑣}) ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
83, 7xchbinx 333 . . . 4 (𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ ¬ (𝑣 ∈ (𝑉 ∖ {𝑣}) ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
92, 8sylibr 233 . . 3 (𝑣𝑉𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒})
10 eqidd 2733 . . . 4 (𝑣𝑉𝑣 = 𝑣)
11 nbgrnself.v . . . . 5 𝑉 = (Vtx‘𝐺)
12 eqid 2732 . . . . 5 (Edg‘𝐺) = (Edg‘𝐺)
1311, 12nbgrval 28582 . . . 4 (𝑣𝑉 → (𝐺 NeighbVtx 𝑣) = {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒})
1410, 13neleq12d 3051 . . 3 (𝑣𝑉 → (𝑣 ∉ (𝐺 NeighbVtx 𝑣) ↔ 𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒}))
159, 14mpbird 256 . 2 (𝑣𝑉𝑣 ∉ (𝐺 NeighbVtx 𝑣))
1615rgen 3063 1 𝑣𝑉 𝑣 ∉ (𝐺 NeighbVtx 𝑣)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1541  wcel 2106  wnel 3046  wral 3061  wrex 3070  {crab 3432  cdif 3944  wss 3947  {csn 4627  {cpr 4629  cfv 6540  (class class class)co 7405  Vtxcvtx 28245  Edgcedg 28296   NeighbVtx cnbgr 28578
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6492  df-fun 6542  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-nbgr 28579
This theorem is referenced by:  nbgrnself2  28606
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