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Theorem sclnbgrisvtx 47721
Description: Every member 𝑋 of the semiclosed neighborhood of a vertex 𝑁 is 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
sclnbgrisvtx (𝑋𝑆𝑋𝑉)
Distinct variable groups:   𝑒,𝑁,𝑛   𝑒,𝑉,𝑛   𝑒,𝐸,𝑛   𝑒,𝑋,𝑛
Allowed substitution hints:   𝑆(𝑒,𝑛)   𝐺(𝑒,𝑛)

Proof of Theorem sclnbgrisvtx
StepHypRef Expression
1 dfsclnbgr2.v . . 3 𝑉 = (Vtx‘𝐺)
2 dfsclnbgr2.s . . 3 𝑆 = {𝑛𝑉 ∣ ∃𝑒𝐸 {𝑁, 𝑛} ⊆ 𝑒}
3 dfsclnbgr2.e . . 3 𝐸 = (Edg‘𝐺)
41, 2, 3sclnbgrel 47719 . 2 (𝑋𝑆 ↔ (𝑋𝑉 ∧ ∃𝑒𝐸 {𝑁, 𝑋} ⊆ 𝑒))
54simplbi 497 1 (𝑋𝑆𝑋𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2108  wrex 3076  {crab 3443  wss 3976  {cpr 4650  cfv 6573  Vtxcvtx 29031  Edgcedg 29082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rex 3077  df-rab 3444  df-v 3490  df-un 3981  df-ss 3993  df-sn 4649  df-pr 4651
This theorem is referenced by: (None)
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