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Theorem nbgrisvtx 29373
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 2735 . . 3 (Edg‘𝐺) = (Edg‘𝐺)
31, 2nbgrel 29372 . 2 (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ ((𝑁𝑉𝐾𝑉) ∧ 𝑁𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒))
4 simp1l 1196 . 2 (((𝑁𝑉𝐾𝑉) ∧ 𝑁𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒) → 𝑁𝑉)
53, 4sylbi 217 1 (𝑁 ∈ (𝐺 NeighbVtx 𝐾) → 𝑁𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wrex 3068  wss 3963  {cpr 4633  cfv 6563  (class class class)co 7431  Vtxcvtx 29028  Edgcedg 29079   NeighbVtx cnbgr 29364
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-nbgr 29365
This theorem is referenced by:  nbgrssvtx  29374  nbgrnself2  29392  nbgrssovtx  29393  frgrnbnb  30322  frgrncvvdeqlem2  30329  frgrncvvdeqlem3  30330  frgrncvvdeqlem9  30336  numclwwlk1lem2foa  30383  numclwwlk1lem2fo  30387
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