Step |
Hyp |
Ref |
Expression |
1 |
|
frgrncvvdeq.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
frgrncvvdeq.d |
⊢ 𝐷 = ( VtxDeg ‘ 𝐺 ) |
3 |
|
frgrwopreglem4a.e |
⊢ 𝐸 = ( Edg ‘ 𝐺 ) |
4 |
|
id |
⊢ ( 𝐺 ∈ FriendGraph → 𝐺 ∈ FriendGraph ) |
5 |
|
simpl |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) → 𝐴 ∈ 𝑉 ) |
6 |
|
simpl |
⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → 𝐵 ∈ 𝑉 ) |
7 |
5 6
|
anim12i |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐵 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ) |
8 |
|
simp2 |
⊢ ( ( ( 𝐷 ‘ 𝐴 ) = ( 𝐷 ‘ 𝑋 ) ∧ ( 𝐷 ‘ 𝐴 ) ≠ ( 𝐷 ‘ 𝐵 ) ∧ ( 𝐷 ‘ 𝑋 ) ≠ ( 𝐷 ‘ 𝑌 ) ) → ( 𝐷 ‘ 𝐴 ) ≠ ( 𝐷 ‘ 𝐵 ) ) |
9 |
1 2 3
|
frgrwopreglem4a |
⊢ ( ( 𝐺 ∈ FriendGraph ∧ ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐷 ‘ 𝐴 ) ≠ ( 𝐷 ‘ 𝐵 ) ) → { 𝐴 , 𝐵 } ∈ 𝐸 ) |
10 |
4 7 8 9
|
syl3an |
⊢ ( ( 𝐺 ∈ FriendGraph ∧ ( ( 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐵 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) ∧ ( ( 𝐷 ‘ 𝐴 ) = ( 𝐷 ‘ 𝑋 ) ∧ ( 𝐷 ‘ 𝐴 ) ≠ ( 𝐷 ‘ 𝐵 ) ∧ ( 𝐷 ‘ 𝑋 ) ≠ ( 𝐷 ‘ 𝑌 ) ) ) → { 𝐴 , 𝐵 } ∈ 𝐸 ) |
11 |
|
simpr |
⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) → 𝑋 ∈ 𝑉 ) |
12 |
11 6
|
anim12ci |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐵 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝐵 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ) |
13 |
|
pm13.18 |
⊢ ( ( ( 𝐷 ‘ 𝐴 ) = ( 𝐷 ‘ 𝑋 ) ∧ ( 𝐷 ‘ 𝐴 ) ≠ ( 𝐷 ‘ 𝐵 ) ) → ( 𝐷 ‘ 𝑋 ) ≠ ( 𝐷 ‘ 𝐵 ) ) |
14 |
13
|
3adant3 |
⊢ ( ( ( 𝐷 ‘ 𝐴 ) = ( 𝐷 ‘ 𝑋 ) ∧ ( 𝐷 ‘ 𝐴 ) ≠ ( 𝐷 ‘ 𝐵 ) ∧ ( 𝐷 ‘ 𝑋 ) ≠ ( 𝐷 ‘ 𝑌 ) ) → ( 𝐷 ‘ 𝑋 ) ≠ ( 𝐷 ‘ 𝐵 ) ) |
15 |
14
|
necomd |
⊢ ( ( ( 𝐷 ‘ 𝐴 ) = ( 𝐷 ‘ 𝑋 ) ∧ ( 𝐷 ‘ 𝐴 ) ≠ ( 𝐷 ‘ 𝐵 ) ∧ ( 𝐷 ‘ 𝑋 ) ≠ ( 𝐷 ‘ 𝑌 ) ) → ( 𝐷 ‘ 𝐵 ) ≠ ( 𝐷 ‘ 𝑋 ) ) |
16 |
1 2 3
|
frgrwopreglem4a |
⊢ ( ( 𝐺 ∈ FriendGraph ∧ ( 𝐵 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐷 ‘ 𝐵 ) ≠ ( 𝐷 ‘ 𝑋 ) ) → { 𝐵 , 𝑋 } ∈ 𝐸 ) |
17 |
4 12 15 16
|
syl3an |
⊢ ( ( 𝐺 ∈ FriendGraph ∧ ( ( 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐵 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) ∧ ( ( 𝐷 ‘ 𝐴 ) = ( 𝐷 ‘ 𝑋 ) ∧ ( 𝐷 ‘ 𝐴 ) ≠ ( 𝐷 ‘ 𝐵 ) ∧ ( 𝐷 ‘ 𝑋 ) ≠ ( 𝐷 ‘ 𝑌 ) ) ) → { 𝐵 , 𝑋 } ∈ 𝐸 ) |
18 |
|
simpr |
⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → 𝑌 ∈ 𝑉 ) |
19 |
11 18
|
anim12i |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐵 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) |
20 |
|
simp3 |
⊢ ( ( ( 𝐷 ‘ 𝐴 ) = ( 𝐷 ‘ 𝑋 ) ∧ ( 𝐷 ‘ 𝐴 ) ≠ ( 𝐷 ‘ 𝐵 ) ∧ ( 𝐷 ‘ 𝑋 ) ≠ ( 𝐷 ‘ 𝑌 ) ) → ( 𝐷 ‘ 𝑋 ) ≠ ( 𝐷 ‘ 𝑌 ) ) |
21 |
1 2 3
|
frgrwopreglem4a |
⊢ ( ( 𝐺 ∈ FriendGraph ∧ ( 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ∧ ( 𝐷 ‘ 𝑋 ) ≠ ( 𝐷 ‘ 𝑌 ) ) → { 𝑋 , 𝑌 } ∈ 𝐸 ) |
22 |
4 19 20 21
|
syl3an |
⊢ ( ( 𝐺 ∈ FriendGraph ∧ ( ( 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐵 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) ∧ ( ( 𝐷 ‘ 𝐴 ) = ( 𝐷 ‘ 𝑋 ) ∧ ( 𝐷 ‘ 𝐴 ) ≠ ( 𝐷 ‘ 𝐵 ) ∧ ( 𝐷 ‘ 𝑋 ) ≠ ( 𝐷 ‘ 𝑌 ) ) ) → { 𝑋 , 𝑌 } ∈ 𝐸 ) |
23 |
5 18
|
anim12ci |
⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐵 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) → ( 𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ) ) |
24 |
|
pm13.181 |
⊢ ( ( ( 𝐷 ‘ 𝐴 ) = ( 𝐷 ‘ 𝑋 ) ∧ ( 𝐷 ‘ 𝑋 ) ≠ ( 𝐷 ‘ 𝑌 ) ) → ( 𝐷 ‘ 𝐴 ) ≠ ( 𝐷 ‘ 𝑌 ) ) |
25 |
24
|
3adant2 |
⊢ ( ( ( 𝐷 ‘ 𝐴 ) = ( 𝐷 ‘ 𝑋 ) ∧ ( 𝐷 ‘ 𝐴 ) ≠ ( 𝐷 ‘ 𝐵 ) ∧ ( 𝐷 ‘ 𝑋 ) ≠ ( 𝐷 ‘ 𝑌 ) ) → ( 𝐷 ‘ 𝐴 ) ≠ ( 𝐷 ‘ 𝑌 ) ) |
26 |
25
|
necomd |
⊢ ( ( ( 𝐷 ‘ 𝐴 ) = ( 𝐷 ‘ 𝑋 ) ∧ ( 𝐷 ‘ 𝐴 ) ≠ ( 𝐷 ‘ 𝐵 ) ∧ ( 𝐷 ‘ 𝑋 ) ≠ ( 𝐷 ‘ 𝑌 ) ) → ( 𝐷 ‘ 𝑌 ) ≠ ( 𝐷 ‘ 𝐴 ) ) |
27 |
1 2 3
|
frgrwopreglem4a |
⊢ ( ( 𝐺 ∈ FriendGraph ∧ ( 𝑌 ∈ 𝑉 ∧ 𝐴 ∈ 𝑉 ) ∧ ( 𝐷 ‘ 𝑌 ) ≠ ( 𝐷 ‘ 𝐴 ) ) → { 𝑌 , 𝐴 } ∈ 𝐸 ) |
28 |
4 23 26 27
|
syl3an |
⊢ ( ( 𝐺 ∈ FriendGraph ∧ ( ( 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐵 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) ∧ ( ( 𝐷 ‘ 𝐴 ) = ( 𝐷 ‘ 𝑋 ) ∧ ( 𝐷 ‘ 𝐴 ) ≠ ( 𝐷 ‘ 𝐵 ) ∧ ( 𝐷 ‘ 𝑋 ) ≠ ( 𝐷 ‘ 𝑌 ) ) ) → { 𝑌 , 𝐴 } ∈ 𝐸 ) |
29 |
22 28
|
jca |
⊢ ( ( 𝐺 ∈ FriendGraph ∧ ( ( 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐵 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) ∧ ( ( 𝐷 ‘ 𝐴 ) = ( 𝐷 ‘ 𝑋 ) ∧ ( 𝐷 ‘ 𝐴 ) ≠ ( 𝐷 ‘ 𝐵 ) ∧ ( 𝐷 ‘ 𝑋 ) ≠ ( 𝐷 ‘ 𝑌 ) ) ) → ( { 𝑋 , 𝑌 } ∈ 𝐸 ∧ { 𝑌 , 𝐴 } ∈ 𝐸 ) ) |
30 |
10 17 29
|
jca31 |
⊢ ( ( 𝐺 ∈ FriendGraph ∧ ( ( 𝐴 ∈ 𝑉 ∧ 𝑋 ∈ 𝑉 ) ∧ ( 𝐵 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) ) ∧ ( ( 𝐷 ‘ 𝐴 ) = ( 𝐷 ‘ 𝑋 ) ∧ ( 𝐷 ‘ 𝐴 ) ≠ ( 𝐷 ‘ 𝐵 ) ∧ ( 𝐷 ‘ 𝑋 ) ≠ ( 𝐷 ‘ 𝑌 ) ) ) → ( ( { 𝐴 , 𝐵 } ∈ 𝐸 ∧ { 𝐵 , 𝑋 } ∈ 𝐸 ) ∧ ( { 𝑋 , 𝑌 } ∈ 𝐸 ∧ { 𝑌 , 𝐴 } ∈ 𝐸 ) ) ) |