Theorem dfnbgrss 48040

Description: Subset chain for different kinds of neighborhoods of a vertex. (Contributed by AV, 16-May-2025.)

Hypotheses

Ref	Expression
dfsclnbgr2.v	⊢ 𝑉 = (Vtx‘𝐺)
dfsclnbgr2.s	⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}
dfsclnbgr2.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
dfnbgrss	⊢ (𝑁 ∈ 𝑉 → ((𝐺 NeighbVtx 𝑁) ⊆ 𝑆 ∧ 𝑆 ⊆ (𝐺 ClNeighbVtx 𝑁)))

Distinct variable groups:   𝑒,𝑁,𝑛   𝑒,𝑉,𝑛   𝑒,𝐸,𝑛   𝑒,𝐺,𝑛

Allowed substitution hints:   𝑆(𝑒,𝑛)

Proof of Theorem dfnbgrss

Step	Hyp	Ref	Expression
1		dfsclnbgr2.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		dfsclnbgr2.s	. . . 4 ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}
3		dfsclnbgr2.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
4	1, 2, 3	dfnbgr5 48039	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = (𝑆 ∖ {𝑁}))
5		difss 4086	. . 3 ⊢ (𝑆 ∖ {𝑁}) ⊆ 𝑆
6	4, 5	eqsstrdi 3976	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) ⊆ 𝑆)
7		ssun2 4129	. . 3 ⊢ 𝑆 ⊆ ({𝑁} ∪ 𝑆)
8	1, 2, 3	dfclnbgr5 48038	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ 𝑆))
9	7, 8	sseqtrrid 3975	. 2 ⊢ (𝑁 ∈ 𝑉 → 𝑆 ⊆ (𝐺 ClNeighbVtx 𝑁))
10	6, 9	jca 511	1 ⊢ (𝑁 ∈ 𝑉 → ((𝐺 NeighbVtx 𝑁) ⊆ 𝑆 ∧ 𝑆 ⊆ (𝐺 ClNeighbVtx 𝑁)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  ∃wrex 3058  {crab 3397   ∖ cdif 3896   ∪ cun 3897   ⊆ wss 3899  {csn 4578  {cpr 4580  ‘cfv 6490  (class class class)co 7356  Vtxcvtx 29018  Edgcedg 29069   NeighbVtx cnbgr 29354   ClNeighbVtx cclnbgr 48006

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 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

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 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-iota 6446  df-fun 6492  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-nbgr 29355  df-clnbgr 48007

This theorem is referenced by:  dfnbgrss2  48047