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Theorem elclnbgrelnbgr 47749
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 2734 . . . . . . 7 (Vtx‘𝐺) = (Vtx‘𝐺)
21clnbgrcl 47745 . . . . . 6 (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → 𝑁 ∈ (Vtx‘𝐺))
31dfclnbgr4 47748 . . . . . 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 2824 . . . 4 (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) ↔ 𝑋 ∈ ({𝑁} ∪ (𝐺 NeighbVtx 𝑁))))
6 elun 4162 . . . . 5 (𝑋 ∈ ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)) ↔ (𝑋 ∈ {𝑁} ∨ 𝑋 ∈ (𝐺 NeighbVtx 𝑁)))
7 elsng 4644 . . . . . . 7 (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ {𝑁} ↔ 𝑋 = 𝑁))
8 eqneqall 2948 . . . . . . . 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 859 . . . . 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 847   = wceq 1536  wcel 2105  wne 2937  cun 3960  {csn 4630  cfv 6562  (class class class)co 7430  Vtxcvtx 29027   NeighbVtx cnbgr 29363   ClNeighbVtx cclnbgr 47742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-nbgr 29364  df-clnbgr 47743
This theorem is referenced by:  isubgr3stgrlem6  47873
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