Theorem elclnbgrelnbgr 47924

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 2731	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	clnbgrcl 47920	. . . . . 6 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → 𝑁 ∈ (Vtx‘𝐺))
3	1	dfclnbgr4 47923	. . . . . 6 ⊢ (𝑁 ∈ (Vtx‘𝐺) → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))
4	2, 3	syl 17	. . . . 5 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)))
5	4	eleq2d 2817	. . . 4 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) ↔ 𝑋 ∈ ({𝑁} ∪ (𝐺 NeighbVtx 𝑁))))
6		elun 4100	. . . . 5 ⊢ (𝑋 ∈ ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)) ↔ (𝑋 ∈ {𝑁} ∨ 𝑋 ∈ (𝐺 NeighbVtx 𝑁)))
7		elsng 4587	. . . . . . 7 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ {𝑁} ↔ 𝑋 = 𝑁))
8		eqneqall 2939	. . . . . . . 8 ⊢ (𝑋 = 𝑁 → (𝑋 ≠ 𝑁 → 𝑋 ∈ (𝐺 NeighbVtx 𝑁)))
9	8	a1i 11	. . . . . . 7 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 = 𝑁 → (𝑋 ≠ 𝑁 → 𝑋 ∈ (𝐺 NeighbVtx 𝑁))))
10	7, 9	sylbid 240	. . . . . 6 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ {𝑁} → (𝑋 ≠ 𝑁 → 𝑋 ∈ (𝐺 NeighbVtx 𝑁))))
11		ax-1 6	. . . . . . 7 ⊢ (𝑋 ∈ (𝐺 NeighbVtx 𝑁) → (𝑋 ≠ 𝑁 → 𝑋 ∈ (𝐺 NeighbVtx 𝑁)))
12	11	a1i 11	. . . . . 6 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ (𝐺 NeighbVtx 𝑁) → (𝑋 ≠ 𝑁 → 𝑋 ∈ (𝐺 NeighbVtx 𝑁))))
13	10, 12	jaod 859	. . . . 5 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → ((𝑋 ∈ {𝑁} ∨ 𝑋 ∈ (𝐺 NeighbVtx 𝑁)) → (𝑋 ≠ 𝑁 → 𝑋 ∈ (𝐺 NeighbVtx 𝑁))))
14	6, 13	biimtrid 242	. . . 4 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)) → (𝑋 ≠ 𝑁 → 𝑋 ∈ (𝐺 NeighbVtx 𝑁))))
15	5, 14	sylbid 240	. . 3 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ≠ 𝑁 → 𝑋 ∈ (𝐺 NeighbVtx 𝑁))))
16	15	pm2.43i 52	. 2 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ≠ 𝑁 → 𝑋 ∈ (𝐺 NeighbVtx 𝑁)))
17	16	imp 406	1 ⊢ ((𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) ∧ 𝑋 ≠ 𝑁) → 𝑋 ∈ (𝐺 NeighbVtx 𝑁))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1541   ∈ wcel 2111   ≠ wne 2928   ∪ cun 3895  {csn 4573  ‘cfv 6481  (class class class)co 7346  Vtxcvtx 28974   NeighbVtx cnbgr 29310   ClNeighbVtx cclnbgr 47917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-nbgr 29311  df-clnbgr 47918

This theorem is referenced by:  isubgr3stgrlem6  48070