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Theorem dfnbgrss2 47845
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 = {𝑛𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒𝐸 (𝑁 = 𝑛𝑒 = {𝑁}))}
dfsclnbgr6.s 𝑆 = {𝑛𝑉 ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒}
Assertion
Ref Expression
dfnbgrss2 (𝑁𝑉 → ((𝐺 NeighbVtx 𝑁) ⊆ 𝑈𝑈𝑆𝑆 ⊆ (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
StepHypRef Expression
1 dfvopnbgr2.v . . . 4 𝑉 = (Vtx‘𝐺)
2 dfvopnbgr2.e . . . 4 𝐸 = (Edg‘𝐺)
3 dfvopnbgr2.u . . . 4 𝑈 = {𝑛𝑉 ∣ (𝑛 ∈ (𝐺 NeighbVtx 𝑁) ∨ ∃𝑒𝐸 (𝑁 = 𝑛𝑒 = {𝑁}))}
41, 2, 3dfnbgr6 47843 . . 3 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = (𝑈 ∖ {𝑁}))
5 difss 4136 . . 3 (𝑈 ∖ {𝑁}) ⊆ 𝑈
64, 5eqsstrdi 4028 . 2 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) ⊆ 𝑈)
7 ssun1 4178 . . 3 𝑈 ⊆ (𝑈 ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒𝐸 𝑛𝑒})
8 dfsclnbgr6.s . . . 4 𝑆 = {𝑛𝑉 ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒}
91, 2, 3, 8dfsclnbgr6 47844 . . 3 (𝑁𝑉𝑆 = (𝑈 ∪ {𝑛 ∈ {𝑁} ∣ ∃𝑒𝐸 𝑛𝑒}))
107, 9sseqtrrid 4027 . 2 (𝑁𝑉𝑈𝑆)
111, 8, 2dfnbgrss 47838 . . 3 (𝑁𝑉 → ((𝐺 NeighbVtx 𝑁) ⊆ 𝑆𝑆 ⊆ (𝐺 ClNeighbVtx 𝑁)))
1211simprd 495 . 2 (𝑁𝑉𝑆 ⊆ (𝐺 ClNeighbVtx 𝑁))
136, 10, 123jca 1129 1 (𝑁𝑉 → ((𝐺 NeighbVtx 𝑁) ⊆ 𝑈𝑈𝑆𝑆 ⊆ (𝐺 ClNeighbVtx 𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 848  w3a 1087   = wceq 1540  wcel 2108  wrex 3070  {crab 3436  cdif 3948  cun 3949  wss 3951  {csn 4626  {cpr 4628  cfv 6561  (class class class)co 7431  Vtxcvtx 29013  Edgcedg 29064   NeighbVtx cnbgr 29349   ClNeighbVtx cclnbgr 47805
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-nbgr 29350  df-clnbgr 47806
This theorem is referenced by: (None)
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