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Theorem nbgrnself2 29377
Description: A class 𝑋 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.) (Revised by AV, 12-Feb-2022.)
Assertion
Ref Expression
nbgrnself2 𝑋 ∉ (𝐺 NeighbVtx 𝑋)

Proof of Theorem nbgrnself2
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . 4 (𝑣 = 𝑋𝑣 = 𝑋)
2 oveq2 7439 . . . 4 (𝑣 = 𝑋 → (𝐺 NeighbVtx 𝑣) = (𝐺 NeighbVtx 𝑋))
31, 2neleq12d 3051 . . 3 (𝑣 = 𝑋 → (𝑣 ∉ (𝐺 NeighbVtx 𝑣) ↔ 𝑋 ∉ (𝐺 NeighbVtx 𝑋)))
4 eqid 2737 . . . 4 (Vtx‘𝐺) = (Vtx‘𝐺)
54nbgrnself 29376 . . 3 𝑣 ∈ (Vtx‘𝐺)𝑣 ∉ (𝐺 NeighbVtx 𝑣)
63, 5vtoclri 3590 . 2 (𝑋 ∈ (Vtx‘𝐺) → 𝑋 ∉ (𝐺 NeighbVtx 𝑋))
74nbgrisvtx 29358 . . . 4 (𝑋 ∈ (𝐺 NeighbVtx 𝑋) → 𝑋 ∈ (Vtx‘𝐺))
87con3i 154 . . 3 𝑋 ∈ (Vtx‘𝐺) → ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑋))
9 df-nel 3047 . . 3 (𝑋 ∉ (𝐺 NeighbVtx 𝑋) ↔ ¬ 𝑋 ∈ (𝐺 NeighbVtx 𝑋))
108, 9sylibr 234 . 2 𝑋 ∈ (Vtx‘𝐺) → 𝑋 ∉ (𝐺 NeighbVtx 𝑋))
116, 10pm2.61i 182 1 𝑋 ∉ (𝐺 NeighbVtx 𝑋)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1540  wcel 2108  wnel 3046  cfv 6561  (class class class)co 7431  Vtxcvtx 29013   NeighbVtx cnbgr 29349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-nbgr 29350
This theorem is referenced by:  nbgrssovtx  29378  nb3grprlem2  29398  isubgr3stgrlem1  47933  isubgr3stgrlem3  47935
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