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Theorem nbgrisvtxOLD 26579
Description: Obsolete version of nbgrisvtx 26577 as of 12-Feb-2022. (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 26-Oct-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
nbgrisvtx.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
nbgrisvtxOLD ((𝐺𝑊𝑁 ∈ (𝐺 NeighbVtx 𝐾)) → 𝑁𝑉)

Proof of Theorem nbgrisvtxOLD
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 nbgrisvtx.v . . . 4 𝑉 = (Vtx‘𝐺)
2 eqid 2799 . . . 4 (Edg‘𝐺) = (Edg‘𝐺)
31, 2nbgrelOLD 26576 . . 3 (𝐺𝑊 → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) ↔ ((𝑁𝑉𝐾𝑉) ∧ 𝑁𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒)))
4 simp1l 1255 . . 3 (((𝑁𝑉𝐾𝑉) ∧ 𝑁𝐾 ∧ ∃𝑒 ∈ (Edg‘𝐺){𝐾, 𝑁} ⊆ 𝑒) → 𝑁𝑉)
53, 4syl6bi 245 . 2 (𝐺𝑊 → (𝑁 ∈ (𝐺 NeighbVtx 𝐾) → 𝑁𝑉))
65imp 396 1 ((𝐺𝑊𝑁 ∈ (𝐺 NeighbVtx 𝐾)) → 𝑁𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385  w3a 1108   = wceq 1653  wcel 2157  wne 2971  wrex 3090  wss 3769  {cpr 4370  cfv 6101  (class class class)co 6878  Vtxcvtx 26231  Edgcedg 26282   NeighbVtx cnbgr 26566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-fal 1667  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fv 6109  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-nbgr 26567
This theorem is referenced by:  nbgrssvtxOLD  26580  nbgrnself2OLD  26601  nbgrssovtxOLD  26602
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