Theorem sclnbgrelself 47828

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 47827	. 2 ⊢ (𝑁 ∈ 𝑆 ↔ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑁} ⊆ 𝑒))
5		dfsn2 4619	. . . . . . 7 ⊢ {𝑁} = {𝑁, 𝑁}
6	5	eqcomi 2745	. . . . . 6 ⊢ {𝑁, 𝑁} = {𝑁}
7	6	sseq1i 3992	. . . . 5 ⊢ ({𝑁, 𝑁} ⊆ 𝑒 ↔ {𝑁} ⊆ 𝑒)
8		snssg 4764	. . . . 5 ⊢ (𝑁 ∈ 𝑉 → (𝑁 ∈ 𝑒 ↔ {𝑁} ⊆ 𝑒))
9	7, 8	bitr4id 290	. . . 4 ⊢ (𝑁 ∈ 𝑉 → ({𝑁, 𝑁} ⊆ 𝑒 ↔ 𝑁 ∈ 𝑒))
10	9	rexbidv 3165	. . 3 ⊢ (𝑁 ∈ 𝑉 → (∃𝑒 ∈ 𝐸 {𝑁, 𝑁} ⊆ 𝑒 ↔ ∃𝑒 ∈ 𝐸 𝑁 ∈ 𝑒))
11	10	pm5.32i 574	. 2 ⊢ ((𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑁} ⊆ 𝑒) ↔ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 𝑁 ∈ 𝑒))
12	4, 11	bitri 275	1 ⊢ (𝑁 ∈ 𝑆 ↔ (𝑁 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 𝑁 ∈ 𝑒))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3061  {crab 3420   ⊆ wss 3931  {csn 4606  {cpr 4608  ‘cfv 6536  Vtxcvtx 28980  Edgcedg 29031

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 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rex 3062  df-rab 3421  df-v 3466  df-un 3936  df-ss 3948  df-sn 4607  df-pr 4609

This theorem is referenced by: (None)