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Theorem nbgrnself2 26146
Description: A class is not a neighbor of itself (whether it is a vertex or not). (Contributed by Alexander van der Vekens, 12-Oct-2017.) (Revised by AV, 3-Nov-2020.)
Assertion
Ref Expression
nbgrnself2 (𝐺𝑊𝑁 ∉ (𝐺 NeighbVtx 𝑁))

Proof of Theorem nbgrnself2
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . . 5 (𝑣 = 𝑁𝑣 = 𝑁)
2 oveq2 6612 . . . . 5 (𝑣 = 𝑁 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑁))
31, 2neleq12d 2897 . . . 4 (𝑣 = 𝑁 → (𝑣 ∉ (𝐺 NeighbVtx 𝑣) ↔ 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
43rspccv 3292 . . 3 (∀𝑣 ∈ (Vtx‘𝐺)𝑣 ∉ (𝐺 NeighbVtx 𝑣) → (𝑁 ∈ (Vtx‘𝐺) → 𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
5 eqid 2621 . . . . 5 (Vtx‘𝐺) = (Vtx‘𝐺)
65nbgrnself 26144 . . . 4 𝑣 ∈ (Vtx‘𝐺)𝑣 ∉ (𝐺 NeighbVtx 𝑣)
76a1i 11 . . 3 (𝐺𝑊 → ∀𝑣 ∈ (Vtx‘𝐺)𝑣 ∉ (𝐺 NeighbVtx 𝑣))
84, 7syl11 33 . 2 (𝑁 ∈ (Vtx‘𝐺) → (𝐺𝑊𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
95nbgrisvtx 26142 . . . . 5 ((𝐺𝑊𝑁 ∈ (𝐺 NeighbVtx 𝑁)) → 𝑁 ∈ (Vtx‘𝐺))
109ex 450 . . . 4 (𝐺𝑊 → (𝑁 ∈ (𝐺 NeighbVtx 𝑁) → 𝑁 ∈ (Vtx‘𝐺)))
1110con3rr3 151 . . 3 𝑁 ∈ (Vtx‘𝐺) → (𝐺𝑊 → ¬ 𝑁 ∈ (𝐺 NeighbVtx 𝑁)))
12 df-nel 2894 . . 3 (𝑁 ∉ (𝐺 NeighbVtx 𝑁) ↔ ¬ 𝑁 ∈ (𝐺 NeighbVtx 𝑁))
1311, 12syl6ibr 242 . 2 𝑁 ∈ (Vtx‘𝐺) → (𝐺𝑊𝑁 ∉ (𝐺 NeighbVtx 𝑁)))
148, 13pm2.61i 176 1 (𝐺𝑊𝑁 ∉ (𝐺 NeighbVtx 𝑁))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1480  wcel 1987  wnel 2893  wral 2907  cfv 5847  (class class class)co 6604  Vtxcvtx 25774   NeighbVtx cnbgr 26111
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 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
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-nel 2894  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-nbgr 26115
This theorem is referenced by:  nbgrssovtx  26147  usgrnbnself2  26149
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