Theorem elclnbgrelnbgr 48452

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 2764	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	clnbgrcl 48448	. . . . . 6 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → 𝑁 ∈ (Vtx‘𝐺))
3	1	dfclnbgr4 48451	. . . . . 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 2850	. . . 4 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) ↔ 𝑋 ∈ ({𝑁} ∪ (𝐺 NeighbVtx 𝑁))))
6		elun 4108	. . . . 5 ⊢ (𝑋 ∈ ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)) ↔ (𝑋 ∈ {𝑁} ∨ 𝑋 ∈ (𝐺 NeighbVtx 𝑁)))
7		elsng 4598	. . . . . . 7 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ {𝑁} ↔ 𝑋 = 𝑁))
8		eqneqall 2970	. . . . . . . 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 242	. . . . . 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 870	. . . . 5 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → ((𝑋 ∈ {𝑁} ∨ 𝑋 ∈ (𝐺 NeighbVtx 𝑁)) → (𝑋 ≠ 𝑁 → 𝑋 ∈ (𝐺 NeighbVtx 𝑁))))
14	6, 13	biimtrid 244	. . . 4 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)) → (𝑋 ≠ 𝑁 → 𝑋 ∈ (𝐺 NeighbVtx 𝑁))))
15	5, 14	sylbid 242	. . 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 410	1 ⊢ ((𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) ∧ 𝑋 ≠ 𝑁) → 𝑋 ∈ (𝐺 NeighbVtx 𝑁))

Syntax hints:   → wi 4   ∧ wa 399   ∨ wo 858   = wceq 1562   ∈ wcel 2144   ≠ wne 2959   ∪ cun 3904  {csn 4584  ‘cfv 6523  (class class class)co 7398  Vtxcvtx 29199   NeighbVtx cnbgr 29535   ClNeighbVtx cclnbgr 48445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-iota 6479  df-fun 6525  df-fv 6531  df-ov 7401  df-oprab 7402  df-mpo 7403  df-1st 7972  df-2nd 7973  df-nbgr 29536  df-clnbgr 48446

This theorem is referenced by:  isubgr3stgrlem6  48598