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Theorem dfnbgrss 47776
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 = {𝑛𝑉 ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒}
dfsclnbgr2.e 𝐸 = (Edg‘𝐺)
Assertion
Ref Expression
dfnbgrss (𝑁𝑉 → ((𝐺 NeighbVtx 𝑁) ⊆ 𝑆𝑆 ⊆ (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
StepHypRef Expression
1 dfsclnbgr2.v . . . 4 𝑉 = (Vtx‘𝐺)
2 dfsclnbgr2.s . . . 4 𝑆 = {𝑛𝑉 ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒}
3 dfsclnbgr2.e . . . 4 𝐸 = (Edg‘𝐺)
41, 2, 3dfnbgr5 47775 . . 3 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) = (𝑆 ∖ {𝑁}))
5 difss 4146 . . 3 (𝑆 ∖ {𝑁}) ⊆ 𝑆
64, 5eqsstrdi 4050 . 2 (𝑁𝑉 → (𝐺 NeighbVtx 𝑁) ⊆ 𝑆)
7 ssun2 4189 . . 3 𝑆 ⊆ ({𝑁} ∪ 𝑆)
81, 2, 3dfclnbgr5 47774 . . 3 (𝑁𝑉 → (𝐺 ClNeighbVtx 𝑁) = ({𝑁} ∪ 𝑆))
97, 8sseqtrrid 4049 . 2 (𝑁𝑉𝑆 ⊆ (𝐺 ClNeighbVtx 𝑁))
106, 9jca 511 1 (𝑁𝑉 → ((𝐺 NeighbVtx 𝑁) ⊆ 𝑆𝑆 ⊆ (𝐺 ClNeighbVtx 𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wcel 2106  wrex 3068  {crab 3433  cdif 3960  cun 3961  wss 3963  {csn 4631  {cpr 4633  cfv 6563  (class class class)co 7431  Vtxcvtx 29028  Edgcedg 29079   NeighbVtx cnbgr 29364   ClNeighbVtx cclnbgr 47743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-nbgr 29365  df-clnbgr 47744
This theorem is referenced by:  dfnbgrss2  47783
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