Step |
Hyp |
Ref |
Expression |
1 |
|
uhgr3cyclex.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
uhgr3cyclex.e |
⊢ 𝐸 = ( Edg ‘ 𝐺 ) |
3 |
|
uhgr3cyclex.i |
⊢ 𝐼 = ( iEdg ‘ 𝐺 ) |
4 |
|
fveq2 |
⊢ ( 𝐽 = 𝐾 → ( 𝐼 ‘ 𝐽 ) = ( 𝐼 ‘ 𝐾 ) ) |
5 |
4
|
eqeq2d |
⊢ ( 𝐽 = 𝐾 → ( { 𝐵 , 𝐶 } = ( 𝐼 ‘ 𝐽 ) ↔ { 𝐵 , 𝐶 } = ( 𝐼 ‘ 𝐾 ) ) ) |
6 |
|
eqeq2 |
⊢ ( ( 𝐼 ‘ 𝐾 ) = { 𝐶 , 𝐴 } → ( { 𝐵 , 𝐶 } = ( 𝐼 ‘ 𝐾 ) ↔ { 𝐵 , 𝐶 } = { 𝐶 , 𝐴 } ) ) |
7 |
6
|
eqcoms |
⊢ ( { 𝐶 , 𝐴 } = ( 𝐼 ‘ 𝐾 ) → ( { 𝐵 , 𝐶 } = ( 𝐼 ‘ 𝐾 ) ↔ { 𝐵 , 𝐶 } = { 𝐶 , 𝐴 } ) ) |
8 |
|
prcom |
⊢ { 𝐶 , 𝐴 } = { 𝐴 , 𝐶 } |
9 |
8
|
eqeq1i |
⊢ ( { 𝐶 , 𝐴 } = { 𝐵 , 𝐶 } ↔ { 𝐴 , 𝐶 } = { 𝐵 , 𝐶 } ) |
10 |
|
simpl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → 𝐴 ∈ 𝑉 ) |
11 |
|
simpr |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → 𝐵 ∈ 𝑉 ) |
12 |
10 11
|
preq1b |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( { 𝐴 , 𝐶 } = { 𝐵 , 𝐶 } ↔ 𝐴 = 𝐵 ) ) |
13 |
12
|
biimpcd |
⊢ ( { 𝐴 , 𝐶 } = { 𝐵 , 𝐶 } → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → 𝐴 = 𝐵 ) ) |
14 |
9 13
|
sylbi |
⊢ ( { 𝐶 , 𝐴 } = { 𝐵 , 𝐶 } → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → 𝐴 = 𝐵 ) ) |
15 |
14
|
eqcoms |
⊢ ( { 𝐵 , 𝐶 } = { 𝐶 , 𝐴 } → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → 𝐴 = 𝐵 ) ) |
16 |
7 15
|
syl6bi |
⊢ ( { 𝐶 , 𝐴 } = ( 𝐼 ‘ 𝐾 ) → ( { 𝐵 , 𝐶 } = ( 𝐼 ‘ 𝐾 ) → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → 𝐴 = 𝐵 ) ) ) |
17 |
16
|
adantl |
⊢ ( ( 𝐾 ∈ dom 𝐼 ∧ { 𝐶 , 𝐴 } = ( 𝐼 ‘ 𝐾 ) ) → ( { 𝐵 , 𝐶 } = ( 𝐼 ‘ 𝐾 ) → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → 𝐴 = 𝐵 ) ) ) |
18 |
17
|
com12 |
⊢ ( { 𝐵 , 𝐶 } = ( 𝐼 ‘ 𝐾 ) → ( ( 𝐾 ∈ dom 𝐼 ∧ { 𝐶 , 𝐴 } = ( 𝐼 ‘ 𝐾 ) ) → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → 𝐴 = 𝐵 ) ) ) |
19 |
5 18
|
syl6bi |
⊢ ( 𝐽 = 𝐾 → ( { 𝐵 , 𝐶 } = ( 𝐼 ‘ 𝐽 ) → ( ( 𝐾 ∈ dom 𝐼 ∧ { 𝐶 , 𝐴 } = ( 𝐼 ‘ 𝐾 ) ) → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → 𝐴 = 𝐵 ) ) ) ) |
20 |
19
|
adantld |
⊢ ( 𝐽 = 𝐾 → ( ( 𝐽 ∈ dom 𝐼 ∧ { 𝐵 , 𝐶 } = ( 𝐼 ‘ 𝐽 ) ) → ( ( 𝐾 ∈ dom 𝐼 ∧ { 𝐶 , 𝐴 } = ( 𝐼 ‘ 𝐾 ) ) → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → 𝐴 = 𝐵 ) ) ) ) |
21 |
20
|
com14 |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ( 𝐽 ∈ dom 𝐼 ∧ { 𝐵 , 𝐶 } = ( 𝐼 ‘ 𝐽 ) ) → ( ( 𝐾 ∈ dom 𝐼 ∧ { 𝐶 , 𝐴 } = ( 𝐼 ‘ 𝐾 ) ) → ( 𝐽 = 𝐾 → 𝐴 = 𝐵 ) ) ) ) |
22 |
21
|
imp32 |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( ( 𝐽 ∈ dom 𝐼 ∧ { 𝐵 , 𝐶 } = ( 𝐼 ‘ 𝐽 ) ) ∧ ( 𝐾 ∈ dom 𝐼 ∧ { 𝐶 , 𝐴 } = ( 𝐼 ‘ 𝐾 ) ) ) ) → ( 𝐽 = 𝐾 → 𝐴 = 𝐵 ) ) |
23 |
22
|
necon3d |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( ( 𝐽 ∈ dom 𝐼 ∧ { 𝐵 , 𝐶 } = ( 𝐼 ‘ 𝐽 ) ) ∧ ( 𝐾 ∈ dom 𝐼 ∧ { 𝐶 , 𝐴 } = ( 𝐼 ‘ 𝐾 ) ) ) ) → ( 𝐴 ≠ 𝐵 → 𝐽 ≠ 𝐾 ) ) |
24 |
23
|
impancom |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝐴 ≠ 𝐵 ) → ( ( ( 𝐽 ∈ dom 𝐼 ∧ { 𝐵 , 𝐶 } = ( 𝐼 ‘ 𝐽 ) ) ∧ ( 𝐾 ∈ dom 𝐼 ∧ { 𝐶 , 𝐴 } = ( 𝐼 ‘ 𝐾 ) ) ) → 𝐽 ≠ 𝐾 ) ) |
25 |
24
|
imp |
⊢ ( ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ 𝐴 ≠ 𝐵 ) ∧ ( ( 𝐽 ∈ dom 𝐼 ∧ { 𝐵 , 𝐶 } = ( 𝐼 ‘ 𝐽 ) ) ∧ ( 𝐾 ∈ dom 𝐼 ∧ { 𝐶 , 𝐴 } = ( 𝐼 ‘ 𝐾 ) ) ) ) → 𝐽 ≠ 𝐾 ) |