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Theorem nbgrnself 27068
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 4718 . . . . 5 (𝑣𝑉 → ¬ 𝑣 ∈ (𝑉 ∖ {𝑣}))
21intnanrd 490 . . . 4 (𝑣𝑉 → ¬ (𝑣 ∈ (𝑉 ∖ {𝑣}) ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
3 df-nel 3121 . . . . 5 (𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ ¬ 𝑣 ∈ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒})
4 preq2 4662 . . . . . . . 8 (𝑛 = 𝑣 → {𝑣, 𝑛} = {𝑣, 𝑣})
54sseq1d 3995 . . . . . . 7 (𝑛 = 𝑣 → ({𝑣, 𝑛} ⊆ 𝑒 ↔ {𝑣, 𝑣} ⊆ 𝑒))
65rexbidv 3294 . . . . . 6 (𝑛 = 𝑣 → (∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
76elrab 3677 . . . . 5 (𝑣 ∈ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ (𝑣 ∈ (𝑉 ∖ {𝑣}) ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
83, 7xchbinx 335 . . . 4 (𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ ¬ (𝑣 ∈ (𝑉 ∖ {𝑣}) ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
92, 8sylibr 235 . . 3 (𝑣𝑉𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒})
10 eqidd 2819 . . . 4 (𝑣𝑉𝑣 = 𝑣)
11 nbgrnself.v . . . . 5 𝑉 = (Vtx‘𝐺)
12 eqid 2818 . . . . 5 (Edg‘𝐺) = (Edg‘𝐺)
1311, 12nbgrval 27045 . . . 4 (𝑣𝑉 → (𝐺 NeighbVtx 𝑣) = {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒})
1410, 13neleq12d 3124 . . 3 (𝑣𝑉 → (𝑣 ∉ (𝐺 NeighbVtx 𝑣) ↔ 𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒}))
159, 14mpbird 258 . 2 (𝑣𝑉𝑣 ∉ (𝐺 NeighbVtx 𝑣))
1615rgen 3145 1 𝑣𝑉 𝑣 ∉ (𝐺 NeighbVtx 𝑣)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1528  wcel 2105  wnel 3120  wral 3135  wrex 3136  {crab 3139  cdif 3930  wss 3933  {csn 4557  {cpr 4559  cfv 6348  (class class class)co 7145  Vtxcvtx 26708  Edgcedg 26759   NeighbVtx cnbgr 27041
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-nbgr 27042
This theorem is referenced by:  nbgrnself2  27069
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