Step |
Hyp |
Ref |
Expression |
1 |
|
umgr2adedgwlk.e |
⊢ 𝐸 = ( Edg ‘ 𝐺 ) |
2 |
1
|
umgredgne |
⊢ ( ( 𝐺 ∈ UMGraph ∧ { 𝐴 , 𝐵 } ∈ 𝐸 ) → 𝐴 ≠ 𝐵 ) |
3 |
2
|
ex |
⊢ ( 𝐺 ∈ UMGraph → ( { 𝐴 , 𝐵 } ∈ 𝐸 → 𝐴 ≠ 𝐵 ) ) |
4 |
1
|
umgredgne |
⊢ ( ( 𝐺 ∈ UMGraph ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) → 𝐵 ≠ 𝐶 ) |
5 |
4
|
ex |
⊢ ( 𝐺 ∈ UMGraph → ( { 𝐵 , 𝐶 } ∈ 𝐸 → 𝐵 ≠ 𝐶 ) ) |
6 |
3 5
|
anim12d |
⊢ ( 𝐺 ∈ UMGraph → ( ( { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) → ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) ) ) |
7 |
6
|
3impib |
⊢ ( ( 𝐺 ∈ UMGraph ∧ { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) → ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) ) |
8 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
9 |
8 1
|
umgrpredgv |
⊢ ( ( 𝐺 ∈ UMGraph ∧ { 𝐴 , 𝐵 } ∈ 𝐸 ) → ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) ) |
10 |
9
|
simpld |
⊢ ( ( 𝐺 ∈ UMGraph ∧ { 𝐴 , 𝐵 } ∈ 𝐸 ) → 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) |
11 |
10
|
3adant3 |
⊢ ( ( 𝐺 ∈ UMGraph ∧ { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) → 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) |
12 |
8 1
|
umgrpredgv |
⊢ ( ( 𝐺 ∈ UMGraph ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) → ( 𝐵 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐶 ∈ ( Vtx ‘ 𝐺 ) ) ) |
13 |
12
|
simpld |
⊢ ( ( 𝐺 ∈ UMGraph ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) → 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) |
14 |
13
|
3adant2 |
⊢ ( ( 𝐺 ∈ UMGraph ∧ { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) → 𝐵 ∈ ( Vtx ‘ 𝐺 ) ) |
15 |
12
|
simprd |
⊢ ( ( 𝐺 ∈ UMGraph ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) → 𝐶 ∈ ( Vtx ‘ 𝐺 ) ) |
16 |
15
|
3adant2 |
⊢ ( ( 𝐺 ∈ UMGraph ∧ { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) → 𝐶 ∈ ( Vtx ‘ 𝐺 ) ) |
17 |
11 14 16
|
3jca |
⊢ ( ( 𝐺 ∈ UMGraph ∧ { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) → ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐶 ∈ ( Vtx ‘ 𝐺 ) ) ) |
18 |
7 17
|
jca |
⊢ ( ( 𝐺 ∈ UMGraph ∧ { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝐶 } ∈ 𝐸 ) → ( ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) ∧ ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐵 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐶 ∈ ( Vtx ‘ 𝐺 ) ) ) ) |