Metamath Proof Explorer

Theorem 2pthnloop

Description: A path of length at least 2 does not contain a loop. In contrast, a path of length 1 can contain/be a loop, see lppthon . (Contributed by AV, 6-Feb-2021)

Ref Expression
Hypothesis 2pthnloop.i ⊢ 𝐼 = ( iEdg ‘ 𝐺 )
Assertion 2pthnloop ( ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ 1 < ( ♯ ‘ 𝐹 ) ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) )

Proof

Step Hyp Ref Expression
1 2pthnloop.i ⊢ 𝐼 = ( iEdg ‘ 𝐺 )
2 pthiswlk ⊢ ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → 𝐹 ( Walks ‘ 𝐺 ) 𝑃 )
3 wlkv ⊢ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) )
4 2 3 syl ⊢ ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) )
5 ispth ⊢ ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) )
6 istrl ⊢ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ Fun ◡ 𝐹 ) )
7 eqid ⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
8 7 1 iswlkg ⊢ ( 𝐺 ∈ V → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) )
9 8 anbi1d ⊢ ( 𝐺 ∈ V → ( ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ∧ Fun ◡ 𝐹 ) ↔ ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ∧ Fun ◡ 𝐹 ) ) )
10 6 9 syl5bb ⊢ ( 𝐺 ∈ V → ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ↔ ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ∧ Fun ◡ 𝐹 ) ) )
11 pthdadjvtx ⊢ ( ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ 1 < ( ♯ ‘ 𝐹 ) ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑃 ‘ 𝑖 ) ≠ ( 𝑃 ‘ ( 𝑖 + 1 ) ) )
12 11 ad5ant245 ⊢ ( ( ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( ( Fun ◡ 𝐹 ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) ∧ 1 < ( ♯ ‘ 𝐹 ) ) ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝑃 ‘ 𝑖 ) ≠ ( 𝑃 ‘ ( 𝑖 + 1 ) ) )
13 12 neneqd ⊢ ( ( ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( ( Fun ◡ 𝐹 ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) ∧ 1 < ( ♯ ‘ 𝐹 ) ) ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ¬ ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) )
14 ifpfal ⊢ ( ¬ ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) → ( if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ↔ { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) )
15 14 adantl ⊢ ( ( ( ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( ( Fun ◡ 𝐹 ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) ∧ 1 < ( ♯ ‘ 𝐹 ) ) ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ¬ ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) → ( if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ↔ { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) )
16 fvexd ⊢ ( ¬ ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) → ( 𝑃 ‘ 𝑖 ) ∈ V )
17 fvexd ⊢ ( ¬ ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) → ( 𝑃 ‘ ( 𝑖 + 1 ) ) ∈ V )
18 neqne ⊢ ( ¬ ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) → ( 𝑃 ‘ 𝑖 ) ≠ ( 𝑃 ‘ ( 𝑖 + 1 ) ) )
19 fvexd ⊢ ( ¬ ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) → ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ∈ V )
20 prsshashgt1 ⊢ ( ( ( ( 𝑃 ‘ 𝑖 ) ∈ V ∧ ( 𝑃 ‘ ( 𝑖 + 1 ) ) ∈ V ∧ ( 𝑃 ‘ 𝑖 ) ≠ ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) ∧ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ∈ V ) → ( { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) → 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) )
21 16 17 18 19 20 syl31anc ⊢ ( ¬ ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) → ( { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) → 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) )
22 21 adantl ⊢ ( ( ( ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( ( Fun ◡ 𝐹 ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) ∧ 1 < ( ♯ ‘ 𝐹 ) ) ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ¬ ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) → ( { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) → 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) )
23 15 22 sylbid ⊢ ( ( ( ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( ( Fun ◡ 𝐹 ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) ∧ 1 < ( ♯ ‘ 𝐹 ) ) ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ¬ ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) ) → ( if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) → 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) )
24 13 23 mpdan ⊢ ( ( ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( ( Fun ◡ 𝐹 ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) ∧ 1 < ( ♯ ‘ 𝐹 ) ) ∧ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) → 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) )
25 24 ralimdva ⊢ ( ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( ( Fun ◡ 𝐹 ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) ∧ 1 < ( ♯ ‘ 𝐹 ) ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) )
26 25 ex ⊢ ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( ( Fun ◡ 𝐹 ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) → ( 1 < ( ♯ ‘ 𝐹 ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) )
27 26 com23 ⊢ ( ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) ∧ 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ) ∧ ( ( Fun ◡ 𝐹 ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) )
28 27 exp31 ⊢ ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( ( ( Fun ◡ 𝐹 ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) ) )
29 28 com24 ⊢ ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ) → ( ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) → ( ( ( Fun ◡ 𝐹 ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) ) )
30 29 3impia ⊢ ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) → ( ( ( Fun ◡ 𝐹 ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) )
31 30 exp4c ⊢ ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) → ( Fun ◡ 𝐹 → ( ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ → ( Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) ) ) )
32 31 imp ⊢ ( ( ( 𝐹 ∈ Word dom 𝐼 ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) if- ( ( 𝑃 ‘ 𝑖 ) = ( 𝑃 ‘ ( 𝑖 + 1 ) ) , ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) = { ( 𝑃 ‘ 𝑖 ) } , { ( 𝑃 ‘ 𝑖 ) , ( 𝑃 ‘ ( 𝑖 + 1 ) ) } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ∧ Fun ◡ 𝐹 ) → ( ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ → ( Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) ) )
33 10 32 syl6bi ⊢ ( 𝐺 ∈ V → ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 → ( ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ → ( Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) ) ) )
34 33 com24 ⊢ ( 𝐺 ∈ V → ( Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ → ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) ) ) )
35 34 com14 ⊢ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 → ( Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ → ( 𝐺 ∈ V → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) ) ) )
36 35 3imp ⊢ ( ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) → ( 𝐺 ∈ V → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) )
37 36 com12 ⊢ ( 𝐺 ∈ V → ( ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ∧ Fun ◡ ( 𝑃 ↾ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ∧ ( ( 𝑃 “ { 0 , ( ♯ ‘ 𝐹 ) } ) ∩ ( 𝑃 “ ( 1 ..^ ( ♯ ‘ 𝐹 ) ) ) ) = ∅ ) → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) )
38 5 37 syl5bi ⊢ ( 𝐺 ∈ V → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) )
39 38 3ad2ant1 ⊢ ( ( 𝐺 ∈ V ∧ 𝐹 ∈ V ∧ 𝑃 ∈ V ) → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) ) )
40 4 39 mpcom ⊢ ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) ) )
41 40 pm2.43i ⊢ ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 → ( 1 < ( ♯ ‘ 𝐹 ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) ) )
42 41 imp ⊢ ( ( 𝐹 ( Paths ‘ 𝐺 ) 𝑃 ∧ 1 < ( ♯ ‘ 𝐹 ) ) → ∀ 𝑖 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) 2 ≤ ( ♯ ‘ ( 𝐼 ‘ ( 𝐹 ‘ 𝑖 ) ) ) )