Theorem sclnbgrelself 48671

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 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}
dfsclnbgr2.e	⊢ 𝐸 = (Edg‘𝐺)

Assertion

Ref	Expression
sclnbgrelself	⊢ (𝑁 ∈ 𝑆 ↔ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 𝑁 ∈ 𝑒))

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

Step	Hyp	Ref	Expression
1		dfsclnbgr2.v	. . 3 ⊢ 𝑉 = (Vtx‘𝐺)
2		dfsclnbgr2.s	. . 3 ⊢ 𝑆 = {𝑛 ∈ 𝑉 ∣ ∃𝑒 ∈ 𝐸 {𝑁, 𝑛} ⊆ 𝑒}
3		dfsclnbgr2.e	. . 3 ⊢ 𝐸 = (Edg‘𝐺)
4	1, 2, 3	sclnbgrel 48670	. 2 ⊢ (𝑁 ∈ 𝑆 ↔ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑁} ⊆ 𝑒))
5		dfsn2 4604	. . . . . . 7 ⊢ {𝑁} = {𝑁, 𝑁}
6	5	eqcomi 2774	. . . . . 6 ⊢ {𝑁, 𝑁} = {𝑁}
7	6	sseq1i 3966	. . . . 5 ⊢ ({𝑁, 𝑁} ⊆ 𝑒 ↔ {𝑁} ⊆ 𝑒)
8		snssg 4751	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ 𝑒 ↔ {𝑁} ⊆ 𝑒))
9	7, 8	bitr4id 293	. . . 4 ⊢ (𝑁 ∈ 𝑉 → ({𝑁, 𝑁} ⊆ 𝑒 ↔ 𝑁 ∈ 𝑒))
10	9	rexbidv 3191	. . 3 ⊢ (𝑁 ∈ 𝑉 → (∃𝑒 ∈ 𝐸 {𝑁, 𝑁} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 𝑁 ∈ 𝑒))
11	10	pm5.32i 585	. 2 ⊢ ((𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑁} ⊆ 𝑒) ↔ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 𝑁 ∈ 𝑒))
12	4, 11	bitri 278	1 ⊢ (𝑁 ∈ 𝑆 ↔ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 𝑁 ∈ 𝑒))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∃wrex 3091  {crab 3418   ⊆ wss 3906  {csn 4591  {cpr 4593  ‘cfv 6540  Vtxcvtx 29401  Edgcedg 29452

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rex 3092  df-rab 3419  df-v 3459  df-un 3911  df-ss 3923  df-sn 4592  df-pr 4594

This theorem is used by: (None)