Step |
Hyp |
Ref |
Expression |
1 |
|
acycgr0v.1 |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
br0 |
⊢ ¬ 𝑓 ∅ 𝑝 |
3 |
|
df-cycls |
⊢ Cycles = ( 𝑔 ∈ V ↦ { 〈 𝑓 , 𝑝 〉 ∣ ( 𝑓 ( Paths ‘ 𝑔 ) 𝑝 ∧ ( 𝑝 ‘ 0 ) = ( 𝑝 ‘ ( ♯ ‘ 𝑓 ) ) ) } ) |
4 |
3
|
relmptopab |
⊢ Rel ( Cycles ‘ 𝐺 ) |
5 |
|
cycliswlk |
⊢ ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 → 𝑓 ( Walks ‘ 𝐺 ) 𝑝 ) |
6 |
|
df-br |
⊢ ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 ↔ 〈 𝑓 , 𝑝 〉 ∈ ( Cycles ‘ 𝐺 ) ) |
7 |
|
df-br |
⊢ ( 𝑓 ( Walks ‘ 𝐺 ) 𝑝 ↔ 〈 𝑓 , 𝑝 〉 ∈ ( Walks ‘ 𝐺 ) ) |
8 |
5 6 7
|
3imtr3i |
⊢ ( 〈 𝑓 , 𝑝 〉 ∈ ( Cycles ‘ 𝐺 ) → 〈 𝑓 , 𝑝 〉 ∈ ( Walks ‘ 𝐺 ) ) |
9 |
4 8
|
relssi |
⊢ ( Cycles ‘ 𝐺 ) ⊆ ( Walks ‘ 𝐺 ) |
10 |
1
|
eqeq1i |
⊢ ( 𝑉 = ∅ ↔ ( Vtx ‘ 𝐺 ) = ∅ ) |
11 |
|
g0wlk0 |
⊢ ( ( Vtx ‘ 𝐺 ) = ∅ → ( Walks ‘ 𝐺 ) = ∅ ) |
12 |
10 11
|
sylbi |
⊢ ( 𝑉 = ∅ → ( Walks ‘ 𝐺 ) = ∅ ) |
13 |
9 12
|
sseqtrid |
⊢ ( 𝑉 = ∅ → ( Cycles ‘ 𝐺 ) ⊆ ∅ ) |
14 |
|
ss0 |
⊢ ( ( Cycles ‘ 𝐺 ) ⊆ ∅ → ( Cycles ‘ 𝐺 ) = ∅ ) |
15 |
|
breq |
⊢ ( ( Cycles ‘ 𝐺 ) = ∅ → ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 ↔ 𝑓 ∅ 𝑝 ) ) |
16 |
15
|
notbid |
⊢ ( ( Cycles ‘ 𝐺 ) = ∅ → ( ¬ 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 ↔ ¬ 𝑓 ∅ 𝑝 ) ) |
17 |
13 14 16
|
3syl |
⊢ ( 𝑉 = ∅ → ( ¬ 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 ↔ ¬ 𝑓 ∅ 𝑝 ) ) |
18 |
2 17
|
mpbiri |
⊢ ( 𝑉 = ∅ → ¬ 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 ) |
19 |
18
|
intnanrd |
⊢ ( 𝑉 = ∅ → ¬ ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 ∧ 𝑓 ≠ ∅ ) ) |
20 |
19
|
nexdv |
⊢ ( 𝑉 = ∅ → ¬ ∃ 𝑝 ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 ∧ 𝑓 ≠ ∅ ) ) |
21 |
20
|
nexdv |
⊢ ( 𝑉 = ∅ → ¬ ∃ 𝑓 ∃ 𝑝 ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 ∧ 𝑓 ≠ ∅ ) ) |
22 |
|
isacycgr |
⊢ ( 𝐺 ∈ 𝑊 → ( 𝐺 ∈ AcyclicGraph ↔ ¬ ∃ 𝑓 ∃ 𝑝 ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 ∧ 𝑓 ≠ ∅ ) ) ) |
23 |
22
|
biimpar |
⊢ ( ( 𝐺 ∈ 𝑊 ∧ ¬ ∃ 𝑓 ∃ 𝑝 ( 𝑓 ( Cycles ‘ 𝐺 ) 𝑝 ∧ 𝑓 ≠ ∅ ) ) → 𝐺 ∈ AcyclicGraph ) |
24 |
21 23
|
sylan2 |
⊢ ( ( 𝐺 ∈ 𝑊 ∧ 𝑉 = ∅ ) → 𝐺 ∈ AcyclicGraph ) |