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Theorem nbgrnself 26238
 Description: A vertex in a graph is not a neighbor of itself. (Contributed by by AV, 3-Nov-2020.) (Revised by AV, 21-Mar-2021.)
Hypothesis
Ref Expression
nbgrisvtx.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
nbgrnself 𝑣𝑉 𝑣 ∉ (𝐺 NeighbVtx 𝑣)
Distinct variable group:   𝑣,𝐺
Allowed substitution hint:   𝑉(𝑣)

Proof of Theorem nbgrnself
Dummy variables 𝑒 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 neldifsnd 4313 . . . . 5 (𝑣𝑉 → ¬ 𝑣 ∈ (𝑉 ∖ {𝑣}))
21intnanrd 962 . . . 4 (𝑣𝑉 → ¬ (𝑣 ∈ (𝑉 ∖ {𝑣}) ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
3 df-nel 2895 . . . . 5 (𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ ¬ 𝑣 ∈ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒})
4 preq2 4260 . . . . . . . 8 (𝑛 = 𝑣 → {𝑣, 𝑛} = {𝑣, 𝑣})
54sseq1d 3624 . . . . . . 7 (𝑛 = 𝑣 → ({𝑣, 𝑛} ⊆ 𝑒 ↔ {𝑣, 𝑣} ⊆ 𝑒))
65rexbidv 3048 . . . . . 6 (𝑛 = 𝑣 → (∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
76elrab 3357 . . . . 5 (𝑣 ∈ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ (𝑣 ∈ (𝑉 ∖ {𝑣}) ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
83, 7xchbinx 324 . . . 4 (𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ ¬ (𝑣 ∈ (𝑉 ∖ {𝑣}) ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
92, 8sylibr 224 . . 3 (𝑣𝑉𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒})
10 eqidd 2621 . . . 4 (𝑣𝑉𝑣 = 𝑣)
11 nbgrisvtx.v . . . . 5 𝑉 = (Vtx‘𝐺)
12 eqid 2620 . . . . 5 (Edg‘𝐺) = (Edg‘𝐺)
1311, 12nbgrval 26215 . . . 4 (𝑣𝑉 → (𝐺 NeighbVtx 𝑣) = {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒})
1410, 13neleq12d 2898 . . 3 (𝑣𝑉 → (𝑣 ∉ (𝐺 NeighbVtx 𝑣) ↔ 𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒}))
159, 14mpbird 247 . 2 (𝑣𝑉𝑣 ∉ (𝐺 NeighbVtx 𝑣))
1615rgen 2919 1 𝑣𝑉 𝑣 ∉ (𝐺 NeighbVtx 𝑣)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 384   = wceq 1481   ∈ wcel 1988   ∉ wnel 2894  ∀wral 2909  ∃wrex 2910  {crab 2913   ∖ cdif 3564   ⊆ wss 3567  {csn 4168  {cpr 4170  ‘cfv 5876  (class class class)co 6635  Vtxcvtx 25855  Edgcedg 25920   NeighbVtx cnbgr 26205 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-nbgr 26209 This theorem is referenced by:  usgrnbnself  26239  nbgrnself2  26240
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