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Theorem sclnbgrelself 47772
Description: A vertex 𝑁 is a member of its semiclosed neighborhood iff there is an edge joining the vertex with 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
sclnbgrelself (𝑁𝑆 ↔ (𝑁𝑉 ∧ ∃𝑒𝐸 𝑁𝑒))
Distinct variable groups:   𝑒,𝑁,𝑛   𝑒,𝑉,𝑛   𝑒,𝐸,𝑛
Allowed substitution hints:   𝑆(𝑒,𝑛)   𝐺(𝑒,𝑛)

Proof of Theorem sclnbgrelself
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 47771 . 2 (𝑁𝑆 ↔ (𝑁𝑉 ∧ ∃𝑒𝐸 {𝑁, 𝑁} ⊆ 𝑒))
5 dfsn2 4644 . . . . . . 7 {𝑁} = {𝑁, 𝑁}
65eqcomi 2744 . . . . . 6 {𝑁, 𝑁} = {𝑁}
76sseq1i 4024 . . . . 5 ({𝑁, 𝑁} ⊆ 𝑒 ↔ {𝑁} ⊆ 𝑒)
8 snssg 4788 . . . . 5 (𝑁𝑉 → (𝑁𝑒 ↔ {𝑁} ⊆ 𝑒))
97, 8bitr4id 290 . . . 4 (𝑁𝑉 → ({𝑁, 𝑁} ⊆ 𝑒𝑁𝑒))
109rexbidv 3177 . . 3 (𝑁𝑉 → (∃𝑒𝐸 {𝑁, 𝑁} ⊆ 𝑒 ↔ ∃𝑒𝐸 𝑁𝑒))
1110pm5.32i 574 . 2 ((𝑁𝑉 ∧ ∃𝑒𝐸 {𝑁, 𝑁} ⊆ 𝑒) ↔ (𝑁𝑉 ∧ ∃𝑒𝐸 𝑁𝑒))
124, 11bitri 275 1 (𝑁𝑆 ↔ (𝑁𝑉 ∧ ∃𝑒𝐸 𝑁𝑒))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  wcel 2106  wrex 3068  {crab 3433  wss 3963  {csn 4631  {cpr 4633  cfv 6563  Vtxcvtx 29028  Edgcedg 29079
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-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rex 3069  df-rab 3434  df-v 3480  df-un 3968  df-ss 3980  df-sn 4632  df-pr 4634
This theorem is referenced by: (None)
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