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Theorem nbgrnself 27303
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 4681 . . . . 5 (𝑣𝑉 → ¬ 𝑣 ∈ (𝑉 ∖ {𝑣}))
21intnanrd 493 . . . 4 (𝑣𝑉 → ¬ (𝑣 ∈ (𝑉 ∖ {𝑣}) ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
3 df-nel 3039 . . . . 5 (𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ ¬ 𝑣 ∈ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒})
4 preq2 4625 . . . . . . . 8 (𝑛 = 𝑣 → {𝑣, 𝑛} = {𝑣, 𝑣})
54sseq1d 3908 . . . . . . 7 (𝑛 = 𝑣 → ({𝑣, 𝑛} ⊆ 𝑒 ↔ {𝑣, 𝑣} ⊆ 𝑒))
65rexbidv 3207 . . . . . 6 (𝑛 = 𝑣 → (∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒 ↔ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
76elrab 3588 . . . . 5 (𝑣 ∈ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ (𝑣 ∈ (𝑉 ∖ {𝑣}) ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
83, 7xchbinx 337 . . . 4 (𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒} ↔ ¬ (𝑣 ∈ (𝑉 ∖ {𝑣}) ∧ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑣} ⊆ 𝑒))
92, 8sylibr 237 . . 3 (𝑣𝑉𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒})
10 eqidd 2739 . . . 4 (𝑣𝑉𝑣 = 𝑣)
11 nbgrnself.v . . . . 5 𝑉 = (Vtx‘𝐺)
12 eqid 2738 . . . . 5 (Edg‘𝐺) = (Edg‘𝐺)
1311, 12nbgrval 27280 . . . 4 (𝑣𝑉 → (𝐺 NeighbVtx 𝑣) = {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒})
1410, 13neleq12d 3042 . . 3 (𝑣𝑉 → (𝑣 ∉ (𝐺 NeighbVtx 𝑣) ↔ 𝑣 ∉ {𝑛 ∈ (𝑉 ∖ {𝑣}) ∣ ∃𝑒 ∈ (Edg‘𝐺){𝑣, 𝑛} ⊆ 𝑒}))
159, 14mpbird 260 . 2 (𝑣𝑉𝑣 ∉ (𝐺 NeighbVtx 𝑣))
1615rgen 3063 1 𝑣𝑉 𝑣 ∉ (𝐺 NeighbVtx 𝑣)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 399   = wceq 1542  wcel 2114  wnel 3038  wral 3053  wrex 3054  {crab 3057  cdif 3840  wss 3843  {csn 4516  {cpr 4518  cfv 6339  (class class class)co 7172  Vtxcvtx 26943  Edgcedg 26994   NeighbVtx cnbgr 27276
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6297  df-fun 6341  df-fv 6347  df-ov 7175  df-oprab 7176  df-mpo 7177  df-nbgr 27277
This theorem is referenced by:  nbgrnself2  27304
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