This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
Metamath Proof Explorer
Description: The closed neighborhood of a vertex K in a graph is a subset of all
vertices of the graph. (Contributed by AV, 9-May-2025)
|
|
Ref |
Expression |
|
Hypothesis |
clnbgrvtxel.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
|
Assertion |
clnbgrssvtx |
⊢ ( 𝐺 ClNeighbVtx 𝐾 ) ⊆ 𝑉 |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
clnbgrvtxel.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
| 2 |
1
|
clnbgrisvtx |
⊢ ( 𝑛 ∈ ( 𝐺 ClNeighbVtx 𝐾 ) → 𝑛 ∈ 𝑉 ) |
| 3 |
2
|
ssriv |
⊢ ( 𝐺 ClNeighbVtx 𝐾 ) ⊆ 𝑉 |