Theorem sclnbgrisvtx 47829

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

Assertion

Ref	Expression
sclnbgrisvtx	⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝑉)

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

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 ⊢ (𝑋 ∈ 𝑆 ↔ (𝑋 ∈ 𝑉 ∧ ∃𝑒 ∈ 𝐸 {𝑁, 𝑋} ⊆ 𝑒))
5	4	simplbi 497	1 ⊢ (𝑋 ∈ 𝑆 → 𝑋 ∈ 𝑉)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ∃wrex 3061  {crab 3420   ⊆ wss 3931  {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)