Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elclnbgrelnbgr Structured version   Visualization version   GIF version

Theorem elclnbgrelnbgr 47812
Description: An element of the closed neighborhood of a vertex which is not the vertex itself is an element of the open neighborhood of the vertex. (Contributed by AV, 24-Sep-2025.)
Assertion
Ref Expression
elclnbgrelnbgr ((𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) ∧ 𝑋𝑁) → 𝑋 ∈ (𝐺 NeighbVtx 𝑁))

Proof of Theorem elclnbgrelnbgr
StepHypRef Expression
1 eqid 2737 . . . . . . 7 (Vtx‘𝐺) = (Vtx‘𝐺)
21clnbgrcl 47808 . . . . . 6 (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → 𝑁 ∈ (Vtx‘𝐺))
31dfclnbgr4 47811 . . . . . 6 (𝑁 ∈ (Vtx‘𝐺) → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))
42, 3syl 17 . . . . 5 (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))
54eleq2d 2827 . . . 4 (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) ↔ 𝑋 ∈ ({𝑁} ∪ (𝐺 NeighbVtx 𝑁))))
6 elun 4153 . . . . 5 (𝑋 ∈ ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)) ↔ (𝑋 ∈ {𝑁} ∨ 𝑋 ∈ (𝐺 NeighbVtx 𝑁)))
7 elsng 4640 . . . . . . 7 (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ {𝑁} ↔ 𝑋 = 𝑁))
8 eqneqall 2951 . . . . . . . 8 (𝑋 = 𝑁 → (𝑋𝑁𝑋 ∈ (𝐺 NeighbVtx 𝑁)))
98a1i 11 . . . . . . 7 (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 = 𝑁 → (𝑋𝑁𝑋 ∈ (𝐺 NeighbVtx 𝑁))))
107, 9sylbid 240 . . . . . 6 (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ {𝑁} → (𝑋𝑁𝑋 ∈ (𝐺 NeighbVtx 𝑁))))
11 ax-1 6 . . . . . . 7 (𝑋 ∈ (𝐺 NeighbVtx 𝑁) → (𝑋𝑁𝑋 ∈ (𝐺 NeighbVtx 𝑁)))
1211a1i 11 . . . . . 6 (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ (𝐺 NeighbVtx 𝑁) → (𝑋𝑁𝑋 ∈ (𝐺 NeighbVtx 𝑁))))
1310, 12jaod 860 . . . . 5 (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → ((𝑋 ∈ {𝑁} ∨ 𝑋 ∈ (𝐺 NeighbVtx 𝑁)) → (𝑋𝑁𝑋 ∈ (𝐺 NeighbVtx 𝑁))))
146, 13biimtrid 242 . . . 4 (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)) → (𝑋𝑁𝑋 ∈ (𝐺 NeighbVtx 𝑁))))
155, 14sylbid 240 . . 3 (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋𝑁𝑋 ∈ (𝐺 NeighbVtx 𝑁))))
1615pm2.43i 52 . 2 (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋𝑁𝑋 ∈ (𝐺 NeighbVtx 𝑁)))
1716imp 406 1 ((𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) ∧ 𝑋𝑁) → 𝑋 ∈ (𝐺 NeighbVtx 𝑁))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 848   = wceq 1540  wcel 2108  wne 2940  cun 3949  {csn 4626  cfv 6561  (class class class)co 7431  Vtxcvtx 29013   NeighbVtx cnbgr 29349   ClNeighbVtx cclnbgr 47805
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-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  df-clnbgr 47806
This theorem is referenced by:  isubgr3stgrlem6  47938
  Copyright terms: Public domain W3C validator