Metamath Proof Explorer

Theorem usgr2wlkneq

Description: The vertices and edges are pairwise different in a walk of length 2 in a simple graph. (Contributed by Alexander van der Vekens, 2-Mar-2018) (Revised by AV, 26-Jan-2021)

Ref Expression
Assertion usgr2wlkneq ( ( ( 𝐺 ∈ USGraph ∧ 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ) ∧ ( ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) )

Proof

Step Hyp Ref Expression
1 usgrupgr ⊢ ( 𝐺 ∈ USGraph → 𝐺 ∈ UPGraph )
2 eqid ⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 )
3 eqid ⊢ ( iEdg ‘ 𝐺 ) = ( iEdg ‘ 𝐺 )
4 2 3 upgriswlk ⊢ ( 𝐺 ∈ UPGraph → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) )
5 1 4 syl ⊢ ( 𝐺 ∈ USGraph → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) ) )
6 2wlklem ⊢ ( ∀ 𝑘 ∈ { 0 , 1 } ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ↔ ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 1 ) ) = { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ) )
7 simplll ⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ 𝑃 : ( 0 ... 2 ) ⟶ ( Vtx ‘ 𝐺 ) ) → 𝐺 ∈ USGraph )
8 fvex ⊢ ( 𝑃 ‘ 0 ) ∈ V
9 3 usgrnloopv ⊢ ( ( 𝐺 ∈ USGraph ∧ ( 𝑃 ‘ 0 ) ∈ V ) → ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ) )
10 7 8 9 sylancl ⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ 𝑃 : ( 0 ... 2 ) ⟶ ( Vtx ‘ 𝐺 ) ) → ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ) )
11 fvex ⊢ ( 𝑃 ‘ 1 ) ∈ V
12 3 usgrnloopv ⊢ ( ( 𝐺 ∈ USGraph ∧ ( 𝑃 ‘ 1 ) ∈ V ) → ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 1 ) ) = { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) )
13 7 11 12 sylancl ⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ 𝑃 : ( 0 ... 2 ) ⟶ ( Vtx ‘ 𝐺 ) ) → ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 1 ) ) = { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) )
14 10 13 anim12d ⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ 𝑃 : ( 0 ... 2 ) ⟶ ( Vtx ‘ 𝐺 ) ) → ( ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 1 ) ) = { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ) → ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ) )
15 fveqeq2 ⊢ ( ( 𝐹 ‘ 0 ) = ( 𝐹 ‘ 1 ) → ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ↔ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 1 ) ) = { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ) )
16 eqtr2 ⊢ ( ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 1 ) ) = { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 1 ) ) = { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ) → { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } = { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } )
17 prcom ⊢ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } = { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 1 ) }
18 17 eqeq2i ⊢ ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } = { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ↔ { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } = { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 1 ) } )
19 fvex ⊢ ( 𝑃 ‘ 2 ) ∈ V
20 8 19 preqr1 ⊢ ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } = { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 1 ) } → ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 2 ) )
21 18 20 sylbi ⊢ ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } = { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } → ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 2 ) )
22 16 21 syl ⊢ ( ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 1 ) ) = { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 1 ) ) = { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ) → ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 2 ) )
23 22 ex ⊢ ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 1 ) ) = { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } → ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 1 ) ) = { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } → ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 2 ) ) )
24 15 23 syl6bi ⊢ ( ( 𝐹 ‘ 0 ) = ( 𝐹 ‘ 1 ) → ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } → ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 1 ) ) = { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } → ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 2 ) ) ) )
25 24 impd ⊢ ( ( 𝐹 ‘ 0 ) = ( 𝐹 ‘ 1 ) → ( ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 1 ) ) = { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ) → ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 2 ) ) )
26 25 com12 ⊢ ( ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 1 ) ) = { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ) → ( ( 𝐹 ‘ 0 ) = ( 𝐹 ‘ 1 ) → ( 𝑃 ‘ 0 ) = ( 𝑃 ‘ 2 ) ) )
27 26 necon3d ⊢ ( ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 1 ) ) = { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ) → ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) → ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) )
28 27 com12 ⊢ ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) → ( ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 1 ) ) = { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ) → ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) )
29 28 adantr ⊢ ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ) → ( ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 1 ) ) = { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ) → ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) )
30 simpl ⊢ ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) )
31 30 adantl ⊢ ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ) → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) )
32 simpl ⊢ ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ) → ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) )
33 simprr ⊢ ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ) → ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) )
34 31 32 33 3jca ⊢ ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ) → ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) )
35 29 34 jctild ⊢ ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ) → ( ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 1 ) ) = { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) ) )
36 35 ex ⊢ ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) → ( ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 1 ) ) = { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) ) ) )
37 36 com23 ⊢ ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) → ( ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 1 ) ) = { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) ) ) )
38 37 adantl ⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ) → ( ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 1 ) ) = { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) ) ) )
39 38 adantr ⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ 𝑃 : ( 0 ... 2 ) ⟶ ( Vtx ‘ 𝐺 ) ) → ( ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 1 ) ) = { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) ) ) )
40 14 39 mpdd ⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ 𝑃 : ( 0 ... 2 ) ⟶ ( Vtx ‘ 𝐺 ) ) → ( ( ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 0 ) ) = { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ∧ ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 1 ) ) = { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) ) )
41 6 40 syl5bi ⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ 𝑃 : ( 0 ... 2 ) ⟶ ( Vtx ‘ 𝐺 ) ) → ( ∀ 𝑘 ∈ { 0 , 1 } ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) ) )
42 41 ex ⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ) → ( 𝑃 : ( 0 ... 2 ) ⟶ ( Vtx ‘ 𝐺 ) → ( ∀ 𝑘 ∈ { 0 , 1 } ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) ) ) )
43 42 com23 ⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ) → ( ∀ 𝑘 ∈ { 0 , 1 } ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } → ( 𝑃 : ( 0 ... 2 ) ⟶ ( Vtx ‘ 𝐺 ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) ) ) )
44 43 ex ⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ) → ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) → ( ∀ 𝑘 ∈ { 0 , 1 } ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } → ( 𝑃 : ( 0 ... 2 ) ⟶ ( Vtx ‘ 𝐺 ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) ) ) ) )
45 fveq2 ⊢ ( ( ♯ ‘ 𝐹 ) = 2 → ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) = ( 𝑃 ‘ 2 ) )
46 45 neeq2d ⊢ ( ( ♯ ‘ 𝐹 ) = 2 → ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ↔ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ) )
47 oveq2 ⊢ ( ( ♯ ‘ 𝐹 ) = 2 → ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = ( 0 ..^ 2 ) )
48 fzo0to2pr ⊢ ( 0 ..^ 2 ) = { 0 , 1 }
49 47 48 eqtrdi ⊢ ( ( ♯ ‘ 𝐹 ) = 2 → ( 0 ..^ ( ♯ ‘ 𝐹 ) ) = { 0 , 1 } )
50 49 raleqdv ⊢ ( ( ♯ ‘ 𝐹 ) = 2 → ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ↔ ∀ 𝑘 ∈ { 0 , 1 } ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) )
51 oveq2 ⊢ ( ( ♯ ‘ 𝐹 ) = 2 → ( 0 ... ( ♯ ‘ 𝐹 ) ) = ( 0 ... 2 ) )
52 51 feq2d ⊢ ( ( ♯ ‘ 𝐹 ) = 2 → ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ↔ 𝑃 : ( 0 ... 2 ) ⟶ ( Vtx ‘ 𝐺 ) ) )
53 52 imbi1d ⊢ ( ( ♯ ‘ 𝐹 ) = 2 → ( ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) ) ↔ ( 𝑃 : ( 0 ... 2 ) ⟶ ( Vtx ‘ 𝐺 ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) ) ) )
54 50 53 imbi12d ⊢ ( ( ♯ ‘ 𝐹 ) = 2 → ( ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } → ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) ) ) ↔ ( ∀ 𝑘 ∈ { 0 , 1 } ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } → ( 𝑃 : ( 0 ... 2 ) ⟶ ( Vtx ‘ 𝐺 ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) ) ) ) )
55 46 54 imbi12d ⊢ ( ( ♯ ‘ 𝐹 ) = 2 → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) → ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } → ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) ) ) ) ↔ ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) → ( ∀ 𝑘 ∈ { 0 , 1 } ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } → ( 𝑃 : ( 0 ... 2 ) ⟶ ( Vtx ‘ 𝐺 ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) ) ) ) ) )
56 44 55 syl5ibrcom ⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ) → ( ( ♯ ‘ 𝐹 ) = 2 → ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) → ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } → ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) ) ) ) ) )
57 56 impd ⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ) → ( ( ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) → ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } → ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) ) ) ) )
58 57 com24 ⊢ ( ( 𝐺 ∈ USGraph ∧ 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ) → ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) → ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } → ( ( ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) ) ) ) )
59 58 ex ⊢ ( 𝐺 ∈ USGraph → ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) → ( 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) → ( ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } → ( ( ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) ) ) ) ) )
60 59 3impd ⊢ ( 𝐺 ∈ USGraph → ( ( 𝐹 ∈ Word dom ( iEdg ‘ 𝐺 ) ∧ 𝑃 : ( 0 ... ( ♯ ‘ 𝐹 ) ) ⟶ ( Vtx ‘ 𝐺 ) ∧ ∀ 𝑘 ∈ ( 0 ..^ ( ♯ ‘ 𝐹 ) ) ( ( iEdg ‘ 𝐺 ) ‘ ( 𝐹 ‘ 𝑘 ) ) = { ( 𝑃 ‘ 𝑘 ) , ( 𝑃 ‘ ( 𝑘 + 1 ) ) } ) → ( ( ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) ) ) )
61 5 60 sylbid ⊢ ( 𝐺 ∈ USGraph → ( 𝐹 ( Walks ‘ 𝐺 ) 𝑃 → ( ( ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) ) ) )
62 61 imp31 ⊢ ( ( ( 𝐺 ∈ USGraph ∧ 𝐹 ( Walks ‘ 𝐺 ) 𝑃 ) ∧ ( ( ♯ ‘ 𝐹 ) = 2 ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ ( ♯ ‘ 𝐹 ) ) ) ) → ( ( ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 1 ) ∧ ( 𝑃 ‘ 0 ) ≠ ( 𝑃 ‘ 2 ) ∧ ( 𝑃 ‘ 1 ) ≠ ( 𝑃 ‘ 2 ) ) ∧ ( 𝐹 ‘ 0 ) ≠ ( 𝐹 ‘ 1 ) ) )