Step |
Hyp |
Ref |
Expression |
1 |
|
iscycl |
⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ) |
2 |
|
relpths |
⊢ Rel ( Paths ‘ 𝐺 ) |
3 |
2
|
brrelex1i |
⊢ ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → 𝐹 ∈ V ) |
4 |
|
hasheq0 |
⊢ ( 𝐹 ∈ V → ( ( ♯ ‘ 𝐹 ) = 0 ↔ 𝐹 = ∅ ) ) |
5 |
4
|
necon3bid |
⊢ ( 𝐹 ∈ V → ( ( ♯ ‘ 𝐹 ) ≠ 0 ↔ 𝐹 ≠ ∅ ) ) |
6 |
5
|
bicomd |
⊢ ( 𝐹 ∈ V → ( 𝐹 ≠ ∅ ↔ ( ♯ ‘ 𝐹 ) ≠ 0 ) ) |
7 |
3 6
|
syl |
⊢ ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 𝐹 ≠ ∅ ↔ ( ♯ ‘ 𝐹 ) ≠ 0 ) ) |
8 |
7
|
biimpa |
⊢ ( ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ 𝐹 ≠ ∅ ) → ( ♯ ‘ 𝐹 ) ≠ 0 ) |
9 |
|
spthdep |
⊢ ( ( 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 ∧ ( ♯ ‘ 𝐹 ) ≠ 0 ) → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) |
10 |
9
|
neneqd |
⊢ ( ( 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 ∧ ( ♯ ‘ 𝐹 ) ≠ 0 ) → ¬ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) |
11 |
10
|
expcom |
⊢ ( ( ♯ ‘ 𝐹 ) ≠ 0 → ( 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 → ¬ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ) |
12 |
8 11
|
syl |
⊢ ( ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ 𝐹 ≠ ∅ ) → ( 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 → ¬ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ) |
13 |
12
|
con2d |
⊢ ( ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ 𝐹 ≠ ∅ ) → ( ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) → ¬ 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 ) ) |
14 |
13
|
impancom |
⊢ ( ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐹 ≠ ∅ → ¬ 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 ) ) |
15 |
1 14
|
sylbi |
⊢ ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → ( 𝐹 ≠ ∅ → ¬ 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 ) ) |
16 |
15
|
com12 |
⊢ ( 𝐹 ≠ ∅ → ( 𝐹 ( Cycles ‘ 𝐺 ) 𝑃 → ¬ 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 ) ) |