Step |
Hyp |
Ref |
Expression |
1 |
|
nbgr2vtx1edg.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
nbgr2vtx1edg.e |
⊢ 𝐸 = ( Edg ‘ 𝐺 ) |
3 |
1
|
fvexi |
⊢ 𝑉 ∈ V |
4 |
|
hash2prb |
⊢ ( 𝑉 ∈ V → ( ( ♯ ‘ 𝑉 ) = 2 ↔ ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ( 𝑎 ≠ 𝑏 ∧ 𝑉 = { 𝑎 , 𝑏 } ) ) ) |
5 |
3 4
|
ax-mp |
⊢ ( ( ♯ ‘ 𝑉 ) = 2 ↔ ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ( 𝑎 ≠ 𝑏 ∧ 𝑉 = { 𝑎 , 𝑏 } ) ) |
6 |
|
simpr |
⊢ ( ( 𝐺 ∈ UHGraph ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) |
7 |
6
|
ancomd |
⊢ ( ( 𝐺 ∈ UHGraph ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → ( 𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉 ) ) |
8 |
7
|
ad2antrr |
⊢ ( ( ( ( 𝐺 ∈ UHGraph ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ 𝑉 = { 𝑎 , 𝑏 } ) ) ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) → ( 𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉 ) ) |
9 |
|
id |
⊢ ( 𝑎 ≠ 𝑏 → 𝑎 ≠ 𝑏 ) |
10 |
9
|
necomd |
⊢ ( 𝑎 ≠ 𝑏 → 𝑏 ≠ 𝑎 ) |
11 |
10
|
adantr |
⊢ ( ( 𝑎 ≠ 𝑏 ∧ 𝑉 = { 𝑎 , 𝑏 } ) → 𝑏 ≠ 𝑎 ) |
12 |
11
|
ad2antlr |
⊢ ( ( ( ( 𝐺 ∈ UHGraph ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ 𝑉 = { 𝑎 , 𝑏 } ) ) ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) → 𝑏 ≠ 𝑎 ) |
13 |
|
prcom |
⊢ { 𝑎 , 𝑏 } = { 𝑏 , 𝑎 } |
14 |
13
|
eleq1i |
⊢ ( { 𝑎 , 𝑏 } ∈ 𝐸 ↔ { 𝑏 , 𝑎 } ∈ 𝐸 ) |
15 |
14
|
biimpi |
⊢ ( { 𝑎 , 𝑏 } ∈ 𝐸 → { 𝑏 , 𝑎 } ∈ 𝐸 ) |
16 |
|
sseq2 |
⊢ ( 𝑒 = { 𝑏 , 𝑎 } → ( { 𝑎 , 𝑏 } ⊆ 𝑒 ↔ { 𝑎 , 𝑏 } ⊆ { 𝑏 , 𝑎 } ) ) |
17 |
16
|
adantl |
⊢ ( ( { 𝑎 , 𝑏 } ∈ 𝐸 ∧ 𝑒 = { 𝑏 , 𝑎 } ) → ( { 𝑎 , 𝑏 } ⊆ 𝑒 ↔ { 𝑎 , 𝑏 } ⊆ { 𝑏 , 𝑎 } ) ) |
18 |
13
|
eqimssi |
⊢ { 𝑎 , 𝑏 } ⊆ { 𝑏 , 𝑎 } |
19 |
18
|
a1i |
⊢ ( { 𝑎 , 𝑏 } ∈ 𝐸 → { 𝑎 , 𝑏 } ⊆ { 𝑏 , 𝑎 } ) |
20 |
15 17 19
|
rspcedvd |
⊢ ( { 𝑎 , 𝑏 } ∈ 𝐸 → ∃ 𝑒 ∈ 𝐸 { 𝑎 , 𝑏 } ⊆ 𝑒 ) |
21 |
20
|
adantl |
⊢ ( ( ( ( 𝐺 ∈ UHGraph ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ 𝑉 = { 𝑎 , 𝑏 } ) ) ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) → ∃ 𝑒 ∈ 𝐸 { 𝑎 , 𝑏 } ⊆ 𝑒 ) |
22 |
1 2
|
nbgrel |
⊢ ( 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ↔ ( ( 𝑏 ∈ 𝑉 ∧ 𝑎 ∈ 𝑉 ) ∧ 𝑏 ≠ 𝑎 ∧ ∃ 𝑒 ∈ 𝐸 { 𝑎 , 𝑏 } ⊆ 𝑒 ) ) |
23 |
8 12 21 22
|
syl3anbrc |
⊢ ( ( ( ( 𝐺 ∈ UHGraph ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ 𝑉 = { 𝑎 , 𝑏 } ) ) ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) → 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ) |
24 |
6
|
ad2antrr |
⊢ ( ( ( ( 𝐺 ∈ UHGraph ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ 𝑉 = { 𝑎 , 𝑏 } ) ) ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) → ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) |
25 |
|
simplrl |
⊢ ( ( ( ( 𝐺 ∈ UHGraph ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ 𝑉 = { 𝑎 , 𝑏 } ) ) ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) → 𝑎 ≠ 𝑏 ) |
26 |
|
id |
⊢ ( { 𝑎 , 𝑏 } ∈ 𝐸 → { 𝑎 , 𝑏 } ∈ 𝐸 ) |
27 |
|
sseq2 |
⊢ ( 𝑒 = { 𝑎 , 𝑏 } → ( { 𝑏 , 𝑎 } ⊆ 𝑒 ↔ { 𝑏 , 𝑎 } ⊆ { 𝑎 , 𝑏 } ) ) |
28 |
27
|
adantl |
⊢ ( ( { 𝑎 , 𝑏 } ∈ 𝐸 ∧ 𝑒 = { 𝑎 , 𝑏 } ) → ( { 𝑏 , 𝑎 } ⊆ 𝑒 ↔ { 𝑏 , 𝑎 } ⊆ { 𝑎 , 𝑏 } ) ) |
29 |
|
prcom |
⊢ { 𝑏 , 𝑎 } = { 𝑎 , 𝑏 } |
30 |
29
|
eqimssi |
⊢ { 𝑏 , 𝑎 } ⊆ { 𝑎 , 𝑏 } |
31 |
30
|
a1i |
⊢ ( { 𝑎 , 𝑏 } ∈ 𝐸 → { 𝑏 , 𝑎 } ⊆ { 𝑎 , 𝑏 } ) |
32 |
26 28 31
|
rspcedvd |
⊢ ( { 𝑎 , 𝑏 } ∈ 𝐸 → ∃ 𝑒 ∈ 𝐸 { 𝑏 , 𝑎 } ⊆ 𝑒 ) |
33 |
32
|
adantl |
⊢ ( ( ( ( 𝐺 ∈ UHGraph ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ 𝑉 = { 𝑎 , 𝑏 } ) ) ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) → ∃ 𝑒 ∈ 𝐸 { 𝑏 , 𝑎 } ⊆ 𝑒 ) |
34 |
1 2
|
nbgrel |
⊢ ( 𝑎 ∈ ( 𝐺 NeighbVtx 𝑏 ) ↔ ( ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ∧ 𝑎 ≠ 𝑏 ∧ ∃ 𝑒 ∈ 𝐸 { 𝑏 , 𝑎 } ⊆ 𝑒 ) ) |
35 |
24 25 33 34
|
syl3anbrc |
⊢ ( ( ( ( 𝐺 ∈ UHGraph ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ 𝑉 = { 𝑎 , 𝑏 } ) ) ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) → 𝑎 ∈ ( 𝐺 NeighbVtx 𝑏 ) ) |
36 |
23 35
|
jca |
⊢ ( ( ( ( 𝐺 ∈ UHGraph ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ 𝑉 = { 𝑎 , 𝑏 } ) ) ∧ { 𝑎 , 𝑏 } ∈ 𝐸 ) → ( 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∧ 𝑎 ∈ ( 𝐺 NeighbVtx 𝑏 ) ) ) |
37 |
36
|
ex |
⊢ ( ( ( 𝐺 ∈ UHGraph ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ 𝑉 = { 𝑎 , 𝑏 } ) ) → ( { 𝑎 , 𝑏 } ∈ 𝐸 → ( 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∧ 𝑎 ∈ ( 𝐺 NeighbVtx 𝑏 ) ) ) ) |
38 |
1 2
|
nbuhgr2vtx1edgblem |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑉 = { 𝑎 , 𝑏 } ∧ 𝑎 ∈ ( 𝐺 NeighbVtx 𝑏 ) ) → { 𝑎 , 𝑏 } ∈ 𝐸 ) |
39 |
38
|
3exp |
⊢ ( 𝐺 ∈ UHGraph → ( 𝑉 = { 𝑎 , 𝑏 } → ( 𝑎 ∈ ( 𝐺 NeighbVtx 𝑏 ) → { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ) |
40 |
39
|
adantr |
⊢ ( ( 𝐺 ∈ UHGraph ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → ( 𝑉 = { 𝑎 , 𝑏 } → ( 𝑎 ∈ ( 𝐺 NeighbVtx 𝑏 ) → { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ) |
41 |
40
|
adantld |
⊢ ( ( 𝐺 ∈ UHGraph ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → ( ( 𝑎 ≠ 𝑏 ∧ 𝑉 = { 𝑎 , 𝑏 } ) → ( 𝑎 ∈ ( 𝐺 NeighbVtx 𝑏 ) → { 𝑎 , 𝑏 } ∈ 𝐸 ) ) ) |
42 |
41
|
imp |
⊢ ( ( ( 𝐺 ∈ UHGraph ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ 𝑉 = { 𝑎 , 𝑏 } ) ) → ( 𝑎 ∈ ( 𝐺 NeighbVtx 𝑏 ) → { 𝑎 , 𝑏 } ∈ 𝐸 ) ) |
43 |
42
|
adantld |
⊢ ( ( ( 𝐺 ∈ UHGraph ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ 𝑉 = { 𝑎 , 𝑏 } ) ) → ( ( 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∧ 𝑎 ∈ ( 𝐺 NeighbVtx 𝑏 ) ) → { 𝑎 , 𝑏 } ∈ 𝐸 ) ) |
44 |
37 43
|
impbid |
⊢ ( ( ( 𝐺 ∈ UHGraph ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ 𝑉 = { 𝑎 , 𝑏 } ) ) → ( { 𝑎 , 𝑏 } ∈ 𝐸 ↔ ( 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∧ 𝑎 ∈ ( 𝐺 NeighbVtx 𝑏 ) ) ) ) |
45 |
|
eleq1 |
⊢ ( 𝑉 = { 𝑎 , 𝑏 } → ( 𝑉 ∈ 𝐸 ↔ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) |
46 |
45
|
adantl |
⊢ ( ( 𝑎 ≠ 𝑏 ∧ 𝑉 = { 𝑎 , 𝑏 } ) → ( 𝑉 ∈ 𝐸 ↔ { 𝑎 , 𝑏 } ∈ 𝐸 ) ) |
47 |
|
id |
⊢ ( 𝑉 = { 𝑎 , 𝑏 } → 𝑉 = { 𝑎 , 𝑏 } ) |
48 |
|
difeq1 |
⊢ ( 𝑉 = { 𝑎 , 𝑏 } → ( 𝑉 ∖ { 𝑣 } ) = ( { 𝑎 , 𝑏 } ∖ { 𝑣 } ) ) |
49 |
48
|
raleqdv |
⊢ ( 𝑉 = { 𝑎 , 𝑏 } → ( ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ↔ ∀ 𝑛 ∈ ( { 𝑎 , 𝑏 } ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ) ) |
50 |
47 49
|
raleqbidv |
⊢ ( 𝑉 = { 𝑎 , 𝑏 } → ( ∀ 𝑣 ∈ 𝑉 ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ↔ ∀ 𝑣 ∈ { 𝑎 , 𝑏 } ∀ 𝑛 ∈ ( { 𝑎 , 𝑏 } ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ) ) |
51 |
|
vex |
⊢ 𝑎 ∈ V |
52 |
|
vex |
⊢ 𝑏 ∈ V |
53 |
|
sneq |
⊢ ( 𝑣 = 𝑎 → { 𝑣 } = { 𝑎 } ) |
54 |
53
|
difeq2d |
⊢ ( 𝑣 = 𝑎 → ( { 𝑎 , 𝑏 } ∖ { 𝑣 } ) = ( { 𝑎 , 𝑏 } ∖ { 𝑎 } ) ) |
55 |
|
oveq2 |
⊢ ( 𝑣 = 𝑎 → ( 𝐺 NeighbVtx 𝑣 ) = ( 𝐺 NeighbVtx 𝑎 ) ) |
56 |
55
|
eleq2d |
⊢ ( 𝑣 = 𝑎 → ( 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ↔ 𝑛 ∈ ( 𝐺 NeighbVtx 𝑎 ) ) ) |
57 |
54 56
|
raleqbidv |
⊢ ( 𝑣 = 𝑎 → ( ∀ 𝑛 ∈ ( { 𝑎 , 𝑏 } ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ↔ ∀ 𝑛 ∈ ( { 𝑎 , 𝑏 } ∖ { 𝑎 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑎 ) ) ) |
58 |
|
sneq |
⊢ ( 𝑣 = 𝑏 → { 𝑣 } = { 𝑏 } ) |
59 |
58
|
difeq2d |
⊢ ( 𝑣 = 𝑏 → ( { 𝑎 , 𝑏 } ∖ { 𝑣 } ) = ( { 𝑎 , 𝑏 } ∖ { 𝑏 } ) ) |
60 |
|
oveq2 |
⊢ ( 𝑣 = 𝑏 → ( 𝐺 NeighbVtx 𝑣 ) = ( 𝐺 NeighbVtx 𝑏 ) ) |
61 |
60
|
eleq2d |
⊢ ( 𝑣 = 𝑏 → ( 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ↔ 𝑛 ∈ ( 𝐺 NeighbVtx 𝑏 ) ) ) |
62 |
59 61
|
raleqbidv |
⊢ ( 𝑣 = 𝑏 → ( ∀ 𝑛 ∈ ( { 𝑎 , 𝑏 } ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ↔ ∀ 𝑛 ∈ ( { 𝑎 , 𝑏 } ∖ { 𝑏 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑏 ) ) ) |
63 |
51 52 57 62
|
ralpr |
⊢ ( ∀ 𝑣 ∈ { 𝑎 , 𝑏 } ∀ 𝑛 ∈ ( { 𝑎 , 𝑏 } ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ↔ ( ∀ 𝑛 ∈ ( { 𝑎 , 𝑏 } ∖ { 𝑎 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∧ ∀ 𝑛 ∈ ( { 𝑎 , 𝑏 } ∖ { 𝑏 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑏 ) ) ) |
64 |
|
difprsn1 |
⊢ ( 𝑎 ≠ 𝑏 → ( { 𝑎 , 𝑏 } ∖ { 𝑎 } ) = { 𝑏 } ) |
65 |
64
|
raleqdv |
⊢ ( 𝑎 ≠ 𝑏 → ( ∀ 𝑛 ∈ ( { 𝑎 , 𝑏 } ∖ { 𝑎 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑎 ) ↔ ∀ 𝑛 ∈ { 𝑏 } 𝑛 ∈ ( 𝐺 NeighbVtx 𝑎 ) ) ) |
66 |
|
eleq1 |
⊢ ( 𝑛 = 𝑏 → ( 𝑛 ∈ ( 𝐺 NeighbVtx 𝑎 ) ↔ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ) ) |
67 |
52 66
|
ralsn |
⊢ ( ∀ 𝑛 ∈ { 𝑏 } 𝑛 ∈ ( 𝐺 NeighbVtx 𝑎 ) ↔ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ) |
68 |
65 67
|
bitrdi |
⊢ ( 𝑎 ≠ 𝑏 → ( ∀ 𝑛 ∈ ( { 𝑎 , 𝑏 } ∖ { 𝑎 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑎 ) ↔ 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ) ) |
69 |
|
difprsn2 |
⊢ ( 𝑎 ≠ 𝑏 → ( { 𝑎 , 𝑏 } ∖ { 𝑏 } ) = { 𝑎 } ) |
70 |
69
|
raleqdv |
⊢ ( 𝑎 ≠ 𝑏 → ( ∀ 𝑛 ∈ ( { 𝑎 , 𝑏 } ∖ { 𝑏 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑏 ) ↔ ∀ 𝑛 ∈ { 𝑎 } 𝑛 ∈ ( 𝐺 NeighbVtx 𝑏 ) ) ) |
71 |
|
eleq1 |
⊢ ( 𝑛 = 𝑎 → ( 𝑛 ∈ ( 𝐺 NeighbVtx 𝑏 ) ↔ 𝑎 ∈ ( 𝐺 NeighbVtx 𝑏 ) ) ) |
72 |
51 71
|
ralsn |
⊢ ( ∀ 𝑛 ∈ { 𝑎 } 𝑛 ∈ ( 𝐺 NeighbVtx 𝑏 ) ↔ 𝑎 ∈ ( 𝐺 NeighbVtx 𝑏 ) ) |
73 |
70 72
|
bitrdi |
⊢ ( 𝑎 ≠ 𝑏 → ( ∀ 𝑛 ∈ ( { 𝑎 , 𝑏 } ∖ { 𝑏 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑏 ) ↔ 𝑎 ∈ ( 𝐺 NeighbVtx 𝑏 ) ) ) |
74 |
68 73
|
anbi12d |
⊢ ( 𝑎 ≠ 𝑏 → ( ( ∀ 𝑛 ∈ ( { 𝑎 , 𝑏 } ∖ { 𝑎 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∧ ∀ 𝑛 ∈ ( { 𝑎 , 𝑏 } ∖ { 𝑏 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑏 ) ) ↔ ( 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∧ 𝑎 ∈ ( 𝐺 NeighbVtx 𝑏 ) ) ) ) |
75 |
63 74
|
syl5bb |
⊢ ( 𝑎 ≠ 𝑏 → ( ∀ 𝑣 ∈ { 𝑎 , 𝑏 } ∀ 𝑛 ∈ ( { 𝑎 , 𝑏 } ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ↔ ( 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∧ 𝑎 ∈ ( 𝐺 NeighbVtx 𝑏 ) ) ) ) |
76 |
50 75
|
sylan9bbr |
⊢ ( ( 𝑎 ≠ 𝑏 ∧ 𝑉 = { 𝑎 , 𝑏 } ) → ( ∀ 𝑣 ∈ 𝑉 ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ↔ ( 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∧ 𝑎 ∈ ( 𝐺 NeighbVtx 𝑏 ) ) ) ) |
77 |
46 76
|
bibi12d |
⊢ ( ( 𝑎 ≠ 𝑏 ∧ 𝑉 = { 𝑎 , 𝑏 } ) → ( ( 𝑉 ∈ 𝐸 ↔ ∀ 𝑣 ∈ 𝑉 ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ) ↔ ( { 𝑎 , 𝑏 } ∈ 𝐸 ↔ ( 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∧ 𝑎 ∈ ( 𝐺 NeighbVtx 𝑏 ) ) ) ) ) |
78 |
77
|
adantl |
⊢ ( ( ( 𝐺 ∈ UHGraph ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ 𝑉 = { 𝑎 , 𝑏 } ) ) → ( ( 𝑉 ∈ 𝐸 ↔ ∀ 𝑣 ∈ 𝑉 ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ) ↔ ( { 𝑎 , 𝑏 } ∈ 𝐸 ↔ ( 𝑏 ∈ ( 𝐺 NeighbVtx 𝑎 ) ∧ 𝑎 ∈ ( 𝐺 NeighbVtx 𝑏 ) ) ) ) ) |
79 |
44 78
|
mpbird |
⊢ ( ( ( 𝐺 ∈ UHGraph ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) ∧ ( 𝑎 ≠ 𝑏 ∧ 𝑉 = { 𝑎 , 𝑏 } ) ) → ( 𝑉 ∈ 𝐸 ↔ ∀ 𝑣 ∈ 𝑉 ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ) ) |
80 |
79
|
ex |
⊢ ( ( 𝐺 ∈ UHGraph ∧ ( 𝑎 ∈ 𝑉 ∧ 𝑏 ∈ 𝑉 ) ) → ( ( 𝑎 ≠ 𝑏 ∧ 𝑉 = { 𝑎 , 𝑏 } ) → ( 𝑉 ∈ 𝐸 ↔ ∀ 𝑣 ∈ 𝑉 ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ) ) ) |
81 |
80
|
rexlimdvva |
⊢ ( 𝐺 ∈ UHGraph → ( ∃ 𝑎 ∈ 𝑉 ∃ 𝑏 ∈ 𝑉 ( 𝑎 ≠ 𝑏 ∧ 𝑉 = { 𝑎 , 𝑏 } ) → ( 𝑉 ∈ 𝐸 ↔ ∀ 𝑣 ∈ 𝑉 ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ) ) ) |
82 |
5 81
|
syl5bi |
⊢ ( 𝐺 ∈ UHGraph → ( ( ♯ ‘ 𝑉 ) = 2 → ( 𝑉 ∈ 𝐸 ↔ ∀ 𝑣 ∈ 𝑉 ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ) ) ) |
83 |
82
|
imp |
⊢ ( ( 𝐺 ∈ UHGraph ∧ ( ♯ ‘ 𝑉 ) = 2 ) → ( 𝑉 ∈ 𝐸 ↔ ∀ 𝑣 ∈ 𝑉 ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑣 } ) 𝑛 ∈ ( 𝐺 NeighbVtx 𝑣 ) ) ) |