Theorem elclnbgrelnbgr 48923

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) ∧ 𝑋 ≠ 𝑁) → 𝑋 ∈ (𝐺 NeighbVtx 𝑁))

Proof of Theorem elclnbgrelnbgr

Step	Hyp	Ref	Expression
1		eqid 2760	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	clnbgrcl 48919	. . . . . 6 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → 𝑁 ∈ (Vtx‘𝐺))
3	1	dfclnbgr4 48922	. . . . . 6 ⊢ (𝑁 ∈ (Vtx‘𝐺) → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))
4	2, 3	syl 18	. . . . 5 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))
5	4	eleq2d 2846	. . . 4 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) ↔ 𝑋 ∈ ({𝑁} ∪ (𝐺 NeighbVtx 𝑁))))
6		elun 4152	. . . . 5 ⊢ (𝑋 ∈ ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)) ↔ (𝑋 ∈ {𝑁} ∨ 𝑋 ∈ (𝐺 NeighbVtx 𝑁)))
7		elsng 4597	. . . . . . 7 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ {𝑁} ↔ 𝑋 = 𝑁))
8		eqneqall 2966	. . . . . . 7 ⊢ (𝑋 = 𝑁 → (𝑋 ≠ 𝑁 → 𝑋 ∈ (𝐺 NeighbVtx 𝑁)))
9	7, 8	biimtrdi 256	. . . . . 6 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ {𝑁} → (𝑋 ≠ 𝑁 → 𝑋 ∈ (𝐺 NeighbVtx 𝑁))))
10		ax1w 13	. . . . . 6 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ (𝐺 NeighbVtx 𝑁) → (𝑋 ≠ 𝑁 → 𝑋 ∈ (𝐺 NeighbVtx 𝑁))))
11	9, 10	jaod 873	. . . . 5 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → ((𝑋 ∈ {𝑁} ∨ 𝑋 ∈ (𝐺 NeighbVtx 𝑁)) → (𝑋 ≠ 𝑁 → 𝑋 ∈ (𝐺 NeighbVtx 𝑁))))
12	6, 11	biimtrid 245	. . . 4 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)) → (𝑋 ≠ 𝑁 → 𝑋 ∈ (𝐺 NeighbVtx 𝑁))))
13	5, 12	sylbid 243	. . 3 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ≠ 𝑁 → 𝑋 ∈ (𝐺 NeighbVtx 𝑁))))
14	13	pm2.43i 53	. 2 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ≠ 𝑁 → 𝑋 ∈ (𝐺 NeighbVtx 𝑁)))
15	14	imp 412	1 ⊢ ((𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) ∧ 𝑋 ≠ 𝑁) → 𝑋 ∈ (𝐺 NeighbVtx 𝑁))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1569   ∈ wcel 2144   ≠ wne 2955   ∪ cun 4149  {csn 4583  ‘cfv 6538  (class class class)co 7420  Vtxcvtx 29574   NeighbVtx cnbgr 29913   ClNeighbVtx cclnbgr 48916

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-nbgr 29914  df-clnbgr 48917

This theorem is used by:  isubgr3stgrlem6  49069