Theorem elclnbgrelnbgr 48618

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 2763	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	clnbgrcl 48614	. . . . . 6 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → 𝑁 ∈ (Vtx‘𝐺))
3	1	dfclnbgr4 48617	. . . . . 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 2849	. . . 4 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) ↔ 𝑋 ∈ ({𝑁} ∪ (𝐺 NeighbVtx 𝑁))))
6		elun 4107	. . . . 5 ⊢ (𝑋 ∈ ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)) ↔ (𝑋 ∈ {𝑁} ∨ 𝑋 ∈ (𝐺 NeighbVtx 𝑁)))
7		elsng 4603	. . . . . . 7 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ {𝑁} ↔ 𝑋 = 𝑁))
8		eqneqall 2969	. . . . . . 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 872	. . . . 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 411	1 ⊢ ((𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) ∧ 𝑋 ≠ 𝑁) → 𝑋 ∈ (𝐺 NeighbVtx 𝑁))

This proof depends on syntax axioms:   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143   ≠ wne 2958   ∪ cun 3903  {csn 4589  ‘cfv 6536  (class class class)co 7410  Vtxcvtx 29355   NeighbVtx cnbgr 29691   ClNeighbVtx cclnbgr 48611

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404  ax-un 7732

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7982  df-2nd 7983  df-nbgr 29692  df-clnbgr 48612

This theorem is used by:  isubgr3stgrlem6  48764