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Theorem nbgrisvtx 28175
Description: Every neighbor 𝑁 of a vertex 𝐾 is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Revised by AV, 12-Feb-2022.)
Hypothesis
Ref Expression
nbgrisvtx.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
nbgrisvtx (𝑁 ∈ (𝐺 NeighbVtx 𝐾) → 𝑁𝑉)

Proof of Theorem nbgrisvtx
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 nbgrisvtx.v . . 3 𝑉 = (Vtx‘𝐺)
2 eqid 2736 . . 3 (Edg‘𝐺) = (Edg‘𝐺)
31, 2nbgrel 28174 . 2 (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ ((𝑁𝑉𝐾𝑉) ∧ 𝑁𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒))
4 simp1l 1197 . 2 (((𝑁𝑉𝐾𝑉) ∧ 𝑁𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒) → 𝑁𝑉)
53, 4sylbi 216 1 (𝑁 ∈ (𝐺 NeighbVtx 𝐾) → 𝑁𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2941  wrex 3071  wss 3908  {cpr 4586  cfv 6493  (class class class)co 7353  Vtxcvtx 27833  Edgcedg 27884   NeighbVtx cnbgr 28166
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 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7668
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fv 6501  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7917  df-2nd 7918  df-nbgr 28167
This theorem is referenced by:  nbgrssvtx  28176  nbgrnself2  28194  nbgrssovtx  28195  frgrnbnb  29123  frgrncvvdeqlem2  29130  frgrncvvdeqlem3  29131  frgrncvvdeqlem9  29137  numclwwlk1lem2foa  29184  numclwwlk1lem2fo  29188
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