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Theorem nbgrisvtx 26176
Description: Every neighbor of a class/vertex is a vertex. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.)
Hypothesis
Ref Expression
nbgrisvtx.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
nbgrisvtx ((𝐺𝑊𝑁 ∈ (𝐺 NeighbVtx 𝐾)) → 𝑁𝑉)

Proof of Theorem nbgrisvtx
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 nbgrisvtx.v . . . 4 𝑉 = (Vtx‘𝐺)
2 eqid 2621 . . . 4 (Edg‘𝐺) = (Edg‘𝐺)
31, 2nbgrel 26159 . . 3 (𝐺𝑊 → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ ((𝑁𝑉𝐾𝑉) ∧ 𝑁𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒)))
4 simp1l 1083 . . 3 (((𝑁𝑉𝐾𝑉) ∧ 𝑁𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒) → 𝑁𝑉)
53, 4syl6bi 243 . 2 (𝐺𝑊 → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) → 𝑁𝑉))
65imp 445 1 ((𝐺𝑊𝑁 ∈ (𝐺 NeighbVtx 𝐾)) → 𝑁𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wrex 2909  wss 3560  {cpr 4157  cfv 5857  (class class class)co 6615  Vtxcvtx 25808  Edgcedg 25873   NeighbVtx cnbgr 26145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-fal 1486  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-nbgr 26149
This theorem is referenced by:  nbgrssvtx  26177  nbgrnself2  26180  nbgrssovtx  26181  frgrnbnb  27055  frgrncvvdeqlem3  27063  frgrncvvdeqlem4  27064  frgrncvvdeqlemC  27070  numclwwlkovf2ex  27109  numclwlk1lem2foa  27113  numclwlk1lem2fo  27117
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