Theorem dfnbgrss2 47863

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

Hypotheses

Ref	Expression
dfvopnbgr2.v	⊢ 𝑉 = (Vtx‘𝐺)
dfvopnbgr2.e	⊢ 𝐸 = (Edg‘𝐺)
dfvopnbgr2.u	⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}
dfsclnbgr6.s	⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}

Assertion

Ref	Expression
dfnbgrss2	⊢ (𝑁 ∈ 𝑉 → ((𝐺 NeighbVtx 𝑁) ⊆ 𝑈 ∧ 𝑈 ⊆ 𝑆 ∧ 𝑆 ⊆ (𝐺 ClNeighbVtx 𝑁)))

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

Allowed substitution hints:   𝑆(𝑒,𝑛)   𝑈(𝑒,𝑛)

Proof of Theorem dfnbgrss2

Step	Hyp	Ref	Expression
1		dfvopnbgr2.v	. . . 4 ⊢ 𝑉 = (Vtx‘𝐺)
2		dfvopnbgr2.e	. . . 4 ⊢ 𝐸 = (Edg‘𝐺)
3		dfvopnbgr2.u	. . . 4 ⊢ 𝑈 = {𝑛 ∈ 𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒 ∈ 𝐸 (𝑁 = 𝑛 ∧ 𝑒 = {𝑁}))}
4	1, 2, 3	dfnbgr6 47861	. . 3 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) = (𝑈 ∖ {𝑁}))
5		difss 4102	. . 3 ⊢ (𝑈 ∖ {𝑁}) ⊆ 𝑈
6	4, 5	eqsstrdi 3994	. 2 ⊢ (𝑁 ∈ 𝑉 → (𝐺 NeighbVtx 𝑁) ⊆ 𝑈)
7		ssun1 4144	. . 3 ⊢ 𝑈 ⊆ (𝑈 ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒})
8		dfsclnbgr6.s	. . . 4 ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}
9	1, 2, 3, 8	dfsclnbgr6 47862	. . 3 ⊢ (𝑁 ∈ 𝑉 → 𝑆 = (𝑈 ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒 ∈ 𝐸 𝑛 ∈ 𝑒}))
10	7, 9	sseqtrrid 3993	. 2 ⊢ (𝑁 ∈ 𝑉 → 𝑈 ⊆ 𝑆)
11	1, 8, 2	dfnbgrss 47856	. . 3 ⊢ (𝑁 ∈ 𝑉 → ((𝐺 NeighbVtx 𝑁) ⊆ 𝑆 ∧ 𝑆 ⊆ (𝐺 ClNeighbVtx 𝑁)))
12	11	simprd 495	. 2 ⊢ (𝑁 ∈ 𝑉 → 𝑆 ⊆ (𝐺 ClNeighbVtx 𝑁))
13	6, 10, 12	3jca 1128	1 ⊢ (𝑁 ∈ 𝑉 → ((𝐺 NeighbVtx 𝑁) ⊆ 𝑈 ∧ 𝑈 ⊆ 𝑆 ∧ 𝑆 ⊆ (𝐺 ClNeighbVtx 𝑁)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∃wrex 3054  {crab 3408   ∖ cdif 3914   ∪ cun 3915   ⊆ wss 3917  {csn 4592  {cpr 4594  ‘cfv 6514  (class class class)co 7390  Vtxcvtx 28930  Edgcedg 28981   NeighbVtx cnbgr 29266   ClNeighbVtx cclnbgr 47823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-1st 7971  df-2nd 7972  df-nbgr 29267  df-clnbgr 47824

This theorem is referenced by: (None)