Theorem elclnbgrelnbgr 48138

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 2737	. . . . . . 7 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺)
2	1	clnbgrcl 48134	. . . . . 6 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → 𝑁 ∈ (Vtx‘𝐺))
3	1	dfclnbgr4 48137	. . . . . 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 2823	. . . 4 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) ↔ 𝑋 ∈ ({𝑁} ∪ (𝐺 NeighbVtx 𝑁))))
6		elun 4106	. . . . 5 ⊢ (𝑋 ∈ ({𝑁} ∪ (𝐺 NeighbVtx 𝑁)) ↔ (𝑋 ∈ {𝑁} ∨ 𝑋 ∈ (𝐺 NeighbVtx 𝑁)))
7		elsng 4595	. . . . . . 7 ⊢ (𝑋 ∈ (𝐺 ClNeighbVtx 𝑁) → (𝑋 ∈ {𝑁} ↔ 𝑋 = 𝑁))
8		eqneqall 2944	. . . . . . . 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 860	. . . . 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 848   = wceq 1542   ∈ wcel 2114   ≠ wne 2933   ∪ cun 3900  {csn 4581  ‘cfv 6493  (class class class)co 7360  Vtxcvtx 29073   NeighbVtx cnbgr 29409   ClNeighbVtx cclnbgr 48131

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6449  df-fun 6495  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-nbgr 29410  df-clnbgr 48132

This theorem is referenced by:  isubgr3stgrlem6  48284