Metamath Proof Explorer
Description: The null graph represented by an empty set is a k-regular simple graph
for every k. (Contributed by AV, 26-Dec-2020)
|
|
Ref |
Expression |
|
Assertion |
0grrusgr |
⊢ ∀ 𝑘 ∈ ℕ0* ∅ RegUSGraph 𝑘 |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
0ex |
⊢ ∅ ∈ V |
2 |
|
vtxval0 |
⊢ ( Vtx ‘ ∅ ) = ∅ |
3 |
|
iedgval0 |
⊢ ( iEdg ‘ ∅ ) = ∅ |
4 |
|
0vtxrusgr |
⊢ ( ( ∅ ∈ V ∧ ( Vtx ‘ ∅ ) = ∅ ∧ ( iEdg ‘ ∅ ) = ∅ ) → ∀ 𝑘 ∈ ℕ0* ∅ RegUSGraph 𝑘 ) |
5 |
1 2 3 4
|
mp3an |
⊢ ∀ 𝑘 ∈ ℕ0* ∅ RegUSGraph 𝑘 |