Step |
Hyp |
Ref |
Expression |
1 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
2 |
|
eqid |
⊢ ( Edg ‘ 𝐺 ) = ( Edg ‘ 𝐺 ) |
3 |
1 2
|
uhgr0vsize0 |
⊢ ( ( 𝐺 ∈ UHGraph ∧ ( ♯ ‘ ( Vtx ‘ 𝐺 ) ) = 0 ) → ( ♯ ‘ ( Edg ‘ 𝐺 ) ) = 0 ) |
4 |
|
fvex |
⊢ ( Edg ‘ 𝐺 ) ∈ V |
5 |
|
hasheq0 |
⊢ ( ( Edg ‘ 𝐺 ) ∈ V → ( ( ♯ ‘ ( Edg ‘ 𝐺 ) ) = 0 ↔ ( Edg ‘ 𝐺 ) = ∅ ) ) |
6 |
4 5
|
ax-mp |
⊢ ( ( ♯ ‘ ( Edg ‘ 𝐺 ) ) = 0 ↔ ( Edg ‘ 𝐺 ) = ∅ ) |
7 |
|
0fin |
⊢ ∅ ∈ Fin |
8 |
|
eleq1 |
⊢ ( ( Edg ‘ 𝐺 ) = ∅ → ( ( Edg ‘ 𝐺 ) ∈ Fin ↔ ∅ ∈ Fin ) ) |
9 |
7 8
|
mpbiri |
⊢ ( ( Edg ‘ 𝐺 ) = ∅ → ( Edg ‘ 𝐺 ) ∈ Fin ) |
10 |
6 9
|
sylbi |
⊢ ( ( ♯ ‘ ( Edg ‘ 𝐺 ) ) = 0 → ( Edg ‘ 𝐺 ) ∈ Fin ) |
11 |
3 10
|
syl |
⊢ ( ( 𝐺 ∈ UHGraph ∧ ( ♯ ‘ ( Vtx ‘ 𝐺 ) ) = 0 ) → ( Edg ‘ 𝐺 ) ∈ Fin ) |