Theorem dfnbgrss 48343

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 48342	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = (𝑆 ∖ {𝑁}))
5		difss 4077	. . 3 ⊢ (𝑆 ∖ {𝑁}) ⊆ 𝑆
6	4, 5	eqsstrdi 3967	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) ⊆ 𝑆)
7		ssun2 4120	. . 3 ⊢ 𝑆 ⊆ ({𝑁} ∪ 𝑆)
8	1, 2, 3	dfclnbgr5 48341	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ 𝑆))
9	7, 8	sseqtrrid 3966	. 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 1542   ∈ wcel 2114  ∃wrex 3062  {crab 3390   ∖ cdif 3887   ∪ cun 3888   ⊆ wss 3890  {csn 4568  {cpr 4570  ‘cfv 6493  (class class class)co 7361  Vtxcvtx 29082  Edgcedg 29133   NeighbVtx cnbgr 29418   ClNeighbVtx cclnbgr 48309

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 5232  ax-nul 5242  ax-pr 5371  ax-un 7683

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 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-iota 6449  df-fun 6495  df-fv 6501  df-ov 7364  df-oprab 7365  df-mpo 7366  df-nbgr 29419  df-clnbgr 48310

This theorem is referenced by:  dfnbgrss2  48350