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Theorem nbgrnself 28884
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 4796 . . . . 5 (𝑣𝑉 → ¬ 𝑣 ∈ (𝑉 ∖ {𝑣}))
21intnanrd 489 . . . 4 (𝑣𝑉 → ¬ (𝑣 ∈ (𝑉 ∖ {𝑣}) ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
3 df-nel 3046 . . . . 5 (𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ ¬ 𝑣 ∈ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒})
4 preq2 4738 . . . . . . . 8 (𝑛 = 𝑣 → {𝑣, 𝑛} = {𝑣, 𝑣})
54sseq1d 4013 . . . . . . 7 (𝑛 = 𝑣 → ({𝑣, 𝑛} ⊆ 𝑒 ↔ {𝑣, 𝑣} ⊆ 𝑒))
65rexbidv 3177 . . . . . 6 (𝑛 = 𝑣 → (∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
76elrab 3683 . . . . 5 (𝑣 ∈ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ (𝑣 ∈ (𝑉 ∖ {𝑣}) ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
83, 7xchbinx 334 . . . 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 2732 . . . 4 (𝑣𝑉𝑣 = 𝑣)
11 nbgrnself.v . . . . 5 𝑉 = (Vtx‘𝐺)
12 eqid 2731 . . . . 5 (Edg‘𝐺) = (Edg‘𝐺)
1311, 12nbgrval 28861 . . . 4 (𝑣𝑉 → (𝐺 NeighbVtx 𝑣) = {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒})
1410, 13neleq12d 3050 . . 3 (𝑣𝑉 → (𝑣 ∉ (𝐺 NeighbVtx 𝑣) ↔ 𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒}))
159, 14mpbird 257 . 2 (𝑣𝑉𝑣 ∉ (𝐺 NeighbVtx 𝑣))
1615rgen 3062 1 𝑣𝑉 𝑣 ∉ (𝐺 NeighbVtx 𝑣)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1540  wcel 2105  wnel 3045  wral 3060  wrex 3069  {crab 3431  cdif 3945  wss 3948  {csn 4628  {cpr 4630  cfv 6543  (class class class)co 7412  Vtxcvtx 28524  Edgcedg 28575   NeighbVtx cnbgr 28857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-nbgr 28858
This theorem is referenced by:  nbgrnself2  28885
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