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Theorem nbgrisvtx 27129
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 2822 . . 3 (Edg‘𝐺) = (Edg‘𝐺)
31, 2nbgrel 27128 . 2 (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ ((𝑁𝑉𝐾𝑉) ∧ 𝑁𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒))
4 simp1l 1194 . 2 (((𝑁𝑉𝐾𝑉) ∧ 𝑁𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒) → 𝑁𝑉)
53, 4sylbi 220 1 (𝑁 ∈ (𝐺 NeighbVtx 𝐾) → 𝑁𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3a 1084   = wceq 1538  wcel 2114  wne 3011  wrex 3131  wss 3908  {cpr 4541  cfv 6334  (class class class)co 7140  Vtxcvtx 26787  Edgcedg 26838   NeighbVtx cnbgr 27120
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-1st 7675  df-2nd 7676  df-nbgr 27121
This theorem is referenced by:  nbgrssvtx  27130  nbgrnself2  27148  nbgrssovtx  27149  frgrnbnb  28076  frgrncvvdeqlem2  28083  frgrncvvdeqlem3  28084  frgrncvvdeqlem9  28090  numclwwlk1lem2foa  28137  numclwwlk1lem2fo  28141
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