Step |
Hyp |
Ref |
Expression |
1 |
|
frgrnbnb.e |
⊢ 𝐸 = ( Edg ‘ 𝐺 ) |
2 |
|
frgrnbnb.n |
⊢ 𝐷 = ( 𝐺 NeighbVtx 𝑋 ) |
3 |
|
frgrusgr |
⊢ ( 𝐺 ∈ FriendGraph → 𝐺 ∈ USGraph ) |
4 |
2
|
eleq2i |
⊢ ( 𝑈 ∈ 𝐷 ↔ 𝑈 ∈ ( 𝐺 NeighbVtx 𝑋 ) ) |
5 |
1
|
nbusgreledg |
⊢ ( 𝐺 ∈ USGraph → ( 𝑈 ∈ ( 𝐺 NeighbVtx 𝑋 ) ↔ { 𝑈 , 𝑋 } ∈ 𝐸 ) ) |
6 |
5
|
biimpd |
⊢ ( 𝐺 ∈ USGraph → ( 𝑈 ∈ ( 𝐺 NeighbVtx 𝑋 ) → { 𝑈 , 𝑋 } ∈ 𝐸 ) ) |
7 |
4 6
|
syl5bi |
⊢ ( 𝐺 ∈ USGraph → ( 𝑈 ∈ 𝐷 → { 𝑈 , 𝑋 } ∈ 𝐸 ) ) |
8 |
2
|
eleq2i |
⊢ ( 𝑊 ∈ 𝐷 ↔ 𝑊 ∈ ( 𝐺 NeighbVtx 𝑋 ) ) |
9 |
1
|
nbusgreledg |
⊢ ( 𝐺 ∈ USGraph → ( 𝑊 ∈ ( 𝐺 NeighbVtx 𝑋 ) ↔ { 𝑊 , 𝑋 } ∈ 𝐸 ) ) |
10 |
9
|
biimpd |
⊢ ( 𝐺 ∈ USGraph → ( 𝑊 ∈ ( 𝐺 NeighbVtx 𝑋 ) → { 𝑊 , 𝑋 } ∈ 𝐸 ) ) |
11 |
8 10
|
syl5bi |
⊢ ( 𝐺 ∈ USGraph → ( 𝑊 ∈ 𝐷 → { 𝑊 , 𝑋 } ∈ 𝐸 ) ) |
12 |
7 11
|
anim12d |
⊢ ( 𝐺 ∈ USGraph → ( ( 𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷 ) → ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) ) ) |
13 |
12
|
imp |
⊢ ( ( 𝐺 ∈ USGraph ∧ ( 𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷 ) ) → ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) ) |
14 |
|
eqid |
⊢ ( Vtx ‘ 𝐺 ) = ( Vtx ‘ 𝐺 ) |
15 |
14
|
nbgrisvtx |
⊢ ( 𝑈 ∈ ( 𝐺 NeighbVtx 𝑋 ) → 𝑈 ∈ ( Vtx ‘ 𝐺 ) ) |
16 |
15 2
|
eleq2s |
⊢ ( 𝑈 ∈ 𝐷 → 𝑈 ∈ ( Vtx ‘ 𝐺 ) ) |
17 |
14
|
nbgrisvtx |
⊢ ( 𝑊 ∈ ( 𝐺 NeighbVtx 𝑋 ) → 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) |
18 |
17 2
|
eleq2s |
⊢ ( 𝑊 ∈ 𝐷 → 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) |
19 |
16 18
|
anim12i |
⊢ ( ( 𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷 ) → ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) |
20 |
19
|
adantl |
⊢ ( ( 𝐺 ∈ USGraph ∧ ( 𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷 ) ) → ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) |
21 |
1 14
|
usgrpredgv |
⊢ ( ( 𝐺 ∈ USGraph ∧ { 𝑈 , 𝐴 } ∈ 𝐸 ) → ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ) |
22 |
21
|
ad2ant2r |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ ( 𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷 ) ) ∧ ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) ) → ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ) |
23 |
|
ax-1 |
⊢ ( 𝐴 = 𝑋 → ( 𝐺 ∈ FriendGraph → 𝐴 = 𝑋 ) ) |
24 |
23
|
2a1d |
⊢ ( 𝐴 = 𝑋 → ( ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) → ( ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) → ( 𝐺 ∈ FriendGraph → 𝐴 = 𝑋 ) ) ) ) |
25 |
24
|
2a1d |
⊢ ( 𝐴 = 𝑋 → ( 𝑈 ≠ 𝑊 → ( ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) → ( ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) → ( ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) → ( 𝐺 ∈ FriendGraph → 𝐴 = 𝑋 ) ) ) ) ) ) |
26 |
|
simpll |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) ) ∧ ( 𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊 ) ) → 𝐺 ∈ USGraph ) |
27 |
|
simprrr |
⊢ ( ( 𝐺 ∈ USGraph ∧ ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) ) → 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) |
28 |
27
|
adantr |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) ) ∧ ( 𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊 ) ) → 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) |
29 |
|
simprrl |
⊢ ( ( 𝐺 ∈ USGraph ∧ ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) ) → 𝑈 ∈ ( Vtx ‘ 𝐺 ) ) |
30 |
29
|
adantr |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) ) ∧ ( 𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊 ) ) → 𝑈 ∈ ( Vtx ‘ 𝐺 ) ) |
31 |
|
necom |
⊢ ( 𝑈 ≠ 𝑊 ↔ 𝑊 ≠ 𝑈 ) |
32 |
31
|
biimpi |
⊢ ( 𝑈 ≠ 𝑊 → 𝑊 ≠ 𝑈 ) |
33 |
32
|
adantl |
⊢ ( ( 𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊 ) → 𝑊 ≠ 𝑈 ) |
34 |
33
|
adantl |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) ) ∧ ( 𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊 ) ) → 𝑊 ≠ 𝑈 ) |
35 |
28 30 34
|
3jca |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) ) ∧ ( 𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊 ) ) → ( 𝑊 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ 𝑈 ) ) |
36 |
|
simprll |
⊢ ( ( 𝐺 ∈ USGraph ∧ ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) ) → 𝑋 ∈ ( Vtx ‘ 𝐺 ) ) |
37 |
36
|
adantr |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) ) ∧ ( 𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊 ) ) → 𝑋 ∈ ( Vtx ‘ 𝐺 ) ) |
38 |
|
simprlr |
⊢ ( ( 𝐺 ∈ USGraph ∧ ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) ) → 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) |
39 |
38
|
adantr |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) ) ∧ ( 𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊 ) ) → 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) |
40 |
|
necom |
⊢ ( 𝐴 ≠ 𝑋 ↔ 𝑋 ≠ 𝐴 ) |
41 |
40
|
biimpi |
⊢ ( 𝐴 ≠ 𝑋 → 𝑋 ≠ 𝐴 ) |
42 |
41
|
adantr |
⊢ ( ( 𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊 ) → 𝑋 ≠ 𝐴 ) |
43 |
42
|
adantl |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) ) ∧ ( 𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊 ) ) → 𝑋 ≠ 𝐴 ) |
44 |
37 39 43
|
3jca |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) ) ∧ ( 𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊 ) ) → ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑋 ≠ 𝐴 ) ) |
45 |
26 35 44
|
3jca |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) ) ∧ ( 𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊 ) ) → ( 𝐺 ∈ USGraph ∧ ( 𝑊 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ 𝑈 ) ∧ ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑋 ≠ 𝐴 ) ) ) |
46 |
45
|
ad4ant14 |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) ) ∧ ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) ) ∧ ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) ) ∧ ( 𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊 ) ) → ( 𝐺 ∈ USGraph ∧ ( 𝑊 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ 𝑈 ) ∧ ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑋 ≠ 𝐴 ) ) ) |
47 |
|
prcom |
⊢ { 𝑈 , 𝑋 } = { 𝑋 , 𝑈 } |
48 |
47
|
eleq1i |
⊢ ( { 𝑈 , 𝑋 } ∈ 𝐸 ↔ { 𝑋 , 𝑈 } ∈ 𝐸 ) |
49 |
48
|
biimpi |
⊢ ( { 𝑈 , 𝑋 } ∈ 𝐸 → { 𝑋 , 𝑈 } ∈ 𝐸 ) |
50 |
49
|
anim1ci |
⊢ ( ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) → ( { 𝑊 , 𝑋 } ∈ 𝐸 ∧ { 𝑋 , 𝑈 } ∈ 𝐸 ) ) |
51 |
50
|
adantl |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) ) ∧ ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) ) → ( { 𝑊 , 𝑋 } ∈ 𝐸 ∧ { 𝑋 , 𝑈 } ∈ 𝐸 ) ) |
52 |
|
prcom |
⊢ { 𝑊 , 𝐴 } = { 𝐴 , 𝑊 } |
53 |
52
|
eleq1i |
⊢ ( { 𝑊 , 𝐴 } ∈ 𝐸 ↔ { 𝐴 , 𝑊 } ∈ 𝐸 ) |
54 |
53
|
biimpi |
⊢ ( { 𝑊 , 𝐴 } ∈ 𝐸 → { 𝐴 , 𝑊 } ∈ 𝐸 ) |
55 |
54
|
anim2i |
⊢ ( ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) → ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝐴 , 𝑊 } ∈ 𝐸 ) ) |
56 |
51 55
|
anim12i |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) ) ∧ ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) ) ∧ ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) ) → ( ( { 𝑊 , 𝑋 } ∈ 𝐸 ∧ { 𝑋 , 𝑈 } ∈ 𝐸 ) ∧ ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝐴 , 𝑊 } ∈ 𝐸 ) ) ) |
57 |
56
|
adantr |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) ) ∧ ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) ) ∧ ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) ) ∧ ( 𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊 ) ) → ( ( { 𝑊 , 𝑋 } ∈ 𝐸 ∧ { 𝑋 , 𝑈 } ∈ 𝐸 ) ∧ ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝐴 , 𝑊 } ∈ 𝐸 ) ) ) |
58 |
14 1
|
4cyclusnfrgr |
⊢ ( ( 𝐺 ∈ USGraph ∧ ( 𝑊 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ≠ 𝑈 ) ∧ ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑋 ≠ 𝐴 ) ) → ( ( ( { 𝑊 , 𝑋 } ∈ 𝐸 ∧ { 𝑋 , 𝑈 } ∈ 𝐸 ) ∧ ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝐴 , 𝑊 } ∈ 𝐸 ) ) → 𝐺 ∉ FriendGraph ) ) |
59 |
46 57 58
|
sylc |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) ) ∧ ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) ) ∧ ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) ) ∧ ( 𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊 ) ) → 𝐺 ∉ FriendGraph ) |
60 |
|
df-nel |
⊢ ( 𝐺 ∉ FriendGraph ↔ ¬ 𝐺 ∈ FriendGraph ) |
61 |
59 60
|
sylib |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) ) ∧ ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) ) ∧ ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) ) ∧ ( 𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊 ) ) → ¬ 𝐺 ∈ FriendGraph ) |
62 |
61
|
pm2.21d |
⊢ ( ( ( ( ( 𝐺 ∈ USGraph ∧ ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) ) ∧ ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) ) ∧ ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) ) ∧ ( 𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊 ) ) → ( 𝐺 ∈ FriendGraph → 𝐴 = 𝑋 ) ) |
63 |
62
|
ex |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) ) ∧ ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) ) ∧ ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) ) → ( ( 𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊 ) → ( 𝐺 ∈ FriendGraph → 𝐴 = 𝑋 ) ) ) |
64 |
63
|
com23 |
⊢ ( ( ( ( 𝐺 ∈ USGraph ∧ ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) ) ∧ ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) ) ∧ ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) ) → ( 𝐺 ∈ FriendGraph → ( ( 𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊 ) → 𝐴 = 𝑋 ) ) ) |
65 |
64
|
exp41 |
⊢ ( 𝐺 ∈ USGraph → ( ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) → ( ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) → ( ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) → ( 𝐺 ∈ FriendGraph → ( ( 𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊 ) → 𝐴 = 𝑋 ) ) ) ) ) ) |
66 |
65
|
com25 |
⊢ ( 𝐺 ∈ USGraph → ( 𝐺 ∈ FriendGraph → ( ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) → ( ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) → ( ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) → ( ( 𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊 ) → 𝐴 = 𝑋 ) ) ) ) ) ) |
67 |
3 66
|
mpcom |
⊢ ( 𝐺 ∈ FriendGraph → ( ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) → ( ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) → ( ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) → ( ( 𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊 ) → 𝐴 = 𝑋 ) ) ) ) ) |
68 |
67
|
com15 |
⊢ ( ( 𝐴 ≠ 𝑋 ∧ 𝑈 ≠ 𝑊 ) → ( ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) → ( ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) → ( ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) → ( 𝐺 ∈ FriendGraph → 𝐴 = 𝑋 ) ) ) ) ) |
69 |
68
|
ex |
⊢ ( 𝐴 ≠ 𝑋 → ( 𝑈 ≠ 𝑊 → ( ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) → ( ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) → ( ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) → ( 𝐺 ∈ FriendGraph → 𝐴 = 𝑋 ) ) ) ) ) ) |
70 |
25 69
|
pm2.61ine |
⊢ ( 𝑈 ≠ 𝑊 → ( ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) → ( ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) → ( ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) → ( 𝐺 ∈ FriendGraph → 𝐴 = 𝑋 ) ) ) ) ) |
71 |
70
|
imp |
⊢ ( ( 𝑈 ≠ 𝑊 ∧ ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) ) → ( ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) → ( ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) → ( 𝐺 ∈ FriendGraph → 𝐴 = 𝑋 ) ) ) ) |
72 |
71
|
com13 |
⊢ ( ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) ∧ ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) ) → ( ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) → ( ( 𝑈 ≠ 𝑊 ∧ ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) ) → ( 𝐺 ∈ FriendGraph → 𝐴 = 𝑋 ) ) ) ) |
73 |
72
|
ex |
⊢ ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) → ( ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) → ( ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) → ( ( 𝑈 ≠ 𝑊 ∧ ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) ) → ( 𝐺 ∈ FriendGraph → 𝐴 = 𝑋 ) ) ) ) ) |
74 |
73
|
com25 |
⊢ ( ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) → ( 𝐺 ∈ FriendGraph → ( ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) → ( ( 𝑈 ≠ 𝑊 ∧ ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) ) → ( ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) → 𝐴 = 𝑋 ) ) ) ) ) |
75 |
74
|
ex |
⊢ ( 𝑋 ∈ ( Vtx ‘ 𝐺 ) → ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) → ( 𝐺 ∈ FriendGraph → ( ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) → ( ( 𝑈 ≠ 𝑊 ∧ ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) ) → ( ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) → 𝐴 = 𝑋 ) ) ) ) ) ) |
76 |
14
|
nbgrcl |
⊢ ( 𝑈 ∈ ( 𝐺 NeighbVtx 𝑋 ) → 𝑋 ∈ ( Vtx ‘ 𝐺 ) ) |
77 |
76 2
|
eleq2s |
⊢ ( 𝑈 ∈ 𝐷 → 𝑋 ∈ ( Vtx ‘ 𝐺 ) ) |
78 |
77
|
adantr |
⊢ ( ( 𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷 ) → 𝑋 ∈ ( Vtx ‘ 𝐺 ) ) |
79 |
78
|
adantl |
⊢ ( ( 𝐺 ∈ USGraph ∧ ( 𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷 ) ) → 𝑋 ∈ ( Vtx ‘ 𝐺 ) ) |
80 |
75 79
|
syl11 |
⊢ ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) → ( ( 𝐺 ∈ USGraph ∧ ( 𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷 ) ) → ( 𝐺 ∈ FriendGraph → ( ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) → ( ( 𝑈 ≠ 𝑊 ∧ ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) ) → ( ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) → 𝐴 = 𝑋 ) ) ) ) ) ) |
81 |
80
|
com34 |
⊢ ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) → ( ( 𝐺 ∈ USGraph ∧ ( 𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷 ) ) → ( ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) → ( 𝐺 ∈ FriendGraph → ( ( 𝑈 ≠ 𝑊 ∧ ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) ) → ( ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) → 𝐴 = 𝑋 ) ) ) ) ) ) |
82 |
81
|
impd |
⊢ ( 𝐴 ∈ ( Vtx ‘ 𝐺 ) → ( ( ( 𝐺 ∈ USGraph ∧ ( 𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷 ) ) ∧ ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) ) → ( 𝐺 ∈ FriendGraph → ( ( 𝑈 ≠ 𝑊 ∧ ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) ) → ( ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) → 𝐴 = 𝑋 ) ) ) ) ) |
83 |
82
|
adantl |
⊢ ( ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝐴 ∈ ( Vtx ‘ 𝐺 ) ) → ( ( ( 𝐺 ∈ USGraph ∧ ( 𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷 ) ) ∧ ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) ) → ( 𝐺 ∈ FriendGraph → ( ( 𝑈 ≠ 𝑊 ∧ ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) ) → ( ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) → 𝐴 = 𝑋 ) ) ) ) ) |
84 |
22 83
|
mpcom |
⊢ ( ( ( 𝐺 ∈ USGraph ∧ ( 𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷 ) ) ∧ ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) ) → ( 𝐺 ∈ FriendGraph → ( ( 𝑈 ≠ 𝑊 ∧ ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) ) → ( ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) → 𝐴 = 𝑋 ) ) ) ) |
85 |
84
|
ex |
⊢ ( ( 𝐺 ∈ USGraph ∧ ( 𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷 ) ) → ( ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) → ( 𝐺 ∈ FriendGraph → ( ( 𝑈 ≠ 𝑊 ∧ ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) ) → ( ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) → 𝐴 = 𝑋 ) ) ) ) ) |
86 |
85
|
com25 |
⊢ ( ( 𝐺 ∈ USGraph ∧ ( 𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷 ) ) → ( ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) → ( 𝐺 ∈ FriendGraph → ( ( 𝑈 ≠ 𝑊 ∧ ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) ) → ( ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) → 𝐴 = 𝑋 ) ) ) ) ) |
87 |
86
|
com14 |
⊢ ( ( 𝑈 ≠ 𝑊 ∧ ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) ) → ( ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) → ( 𝐺 ∈ FriendGraph → ( ( 𝐺 ∈ USGraph ∧ ( 𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷 ) ) → ( ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) → 𝐴 = 𝑋 ) ) ) ) ) |
88 |
87
|
ex |
⊢ ( 𝑈 ≠ 𝑊 → ( ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) → ( ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) → ( 𝐺 ∈ FriendGraph → ( ( 𝐺 ∈ USGraph ∧ ( 𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷 ) ) → ( ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) → 𝐴 = 𝑋 ) ) ) ) ) ) |
89 |
88
|
com15 |
⊢ ( ( 𝐺 ∈ USGraph ∧ ( 𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷 ) ) → ( ( { 𝑈 , 𝑋 } ∈ 𝐸 ∧ { 𝑊 , 𝑋 } ∈ 𝐸 ) → ( ( 𝑈 ∈ ( Vtx ‘ 𝐺 ) ∧ 𝑊 ∈ ( Vtx ‘ 𝐺 ) ) → ( 𝐺 ∈ FriendGraph → ( 𝑈 ≠ 𝑊 → ( ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) → 𝐴 = 𝑋 ) ) ) ) ) ) |
90 |
13 20 89
|
mp2d |
⊢ ( ( 𝐺 ∈ USGraph ∧ ( 𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷 ) ) → ( 𝐺 ∈ FriendGraph → ( 𝑈 ≠ 𝑊 → ( ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) → 𝐴 = 𝑋 ) ) ) ) |
91 |
90
|
ex |
⊢ ( 𝐺 ∈ USGraph → ( ( 𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷 ) → ( 𝐺 ∈ FriendGraph → ( 𝑈 ≠ 𝑊 → ( ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) → 𝐴 = 𝑋 ) ) ) ) ) |
92 |
91
|
com23 |
⊢ ( 𝐺 ∈ USGraph → ( 𝐺 ∈ FriendGraph → ( ( 𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷 ) → ( 𝑈 ≠ 𝑊 → ( ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) → 𝐴 = 𝑋 ) ) ) ) ) |
93 |
3 92
|
mpcom |
⊢ ( 𝐺 ∈ FriendGraph → ( ( 𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷 ) → ( 𝑈 ≠ 𝑊 → ( ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) → 𝐴 = 𝑋 ) ) ) ) |
94 |
93
|
3imp |
⊢ ( ( 𝐺 ∈ FriendGraph ∧ ( 𝑈 ∈ 𝐷 ∧ 𝑊 ∈ 𝐷 ) ∧ 𝑈 ≠ 𝑊 ) → ( ( { 𝑈 , 𝐴 } ∈ 𝐸 ∧ { 𝑊 , 𝐴 } ∈ 𝐸 ) → 𝐴 = 𝑋 ) ) |