Step |
Hyp |
Ref |
Expression |
1 |
|
nbuhgr.v |
⊢ 𝑉 = ( Vtx ‘ 𝐺 ) |
2 |
|
nbuhgr.e |
⊢ 𝐸 = ( Edg ‘ 𝐺 ) |
3 |
1 2
|
nbgrval |
⊢ ( 𝑁 ∈ 𝑉 → ( 𝐺 NeighbVtx 𝑁 ) = { 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ∣ ∃ 𝑒 ∈ 𝐸 { 𝑁 , 𝑛 } ⊆ 𝑒 } ) |
4 |
3
|
a1d |
⊢ ( 𝑁 ∈ 𝑉 → ( ( 𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋 ) → ( 𝐺 NeighbVtx 𝑁 ) = { 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ∣ ∃ 𝑒 ∈ 𝐸 { 𝑁 , 𝑛 } ⊆ 𝑒 } ) ) |
5 |
|
df-nel |
⊢ ( 𝑁 ∉ 𝑉 ↔ ¬ 𝑁 ∈ 𝑉 ) |
6 |
1
|
nbgrnvtx0 |
⊢ ( 𝑁 ∉ 𝑉 → ( 𝐺 NeighbVtx 𝑁 ) = ∅ ) |
7 |
5 6
|
sylbir |
⊢ ( ¬ 𝑁 ∈ 𝑉 → ( 𝐺 NeighbVtx 𝑁 ) = ∅ ) |
8 |
7
|
adantr |
⊢ ( ( ¬ 𝑁 ∈ 𝑉 ∧ ( 𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋 ) ) → ( 𝐺 NeighbVtx 𝑁 ) = ∅ ) |
9 |
|
simpl |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋 ) → 𝐺 ∈ UHGraph ) |
10 |
9
|
adantr |
⊢ ( ( ( 𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋 ) ∧ 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ) → 𝐺 ∈ UHGraph ) |
11 |
2
|
eleq2i |
⊢ ( 𝑒 ∈ 𝐸 ↔ 𝑒 ∈ ( Edg ‘ 𝐺 ) ) |
12 |
11
|
biimpi |
⊢ ( 𝑒 ∈ 𝐸 → 𝑒 ∈ ( Edg ‘ 𝐺 ) ) |
13 |
|
edguhgr |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑒 ∈ ( Edg ‘ 𝐺 ) ) → 𝑒 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ) |
14 |
10 12 13
|
syl2an |
⊢ ( ( ( ( 𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋 ) ∧ 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ) ∧ 𝑒 ∈ 𝐸 ) → 𝑒 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ) |
15 |
|
velpw |
⊢ ( 𝑒 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ↔ 𝑒 ⊆ ( Vtx ‘ 𝐺 ) ) |
16 |
1
|
eqcomi |
⊢ ( Vtx ‘ 𝐺 ) = 𝑉 |
17 |
16
|
sseq2i |
⊢ ( 𝑒 ⊆ ( Vtx ‘ 𝐺 ) ↔ 𝑒 ⊆ 𝑉 ) |
18 |
15 17
|
bitri |
⊢ ( 𝑒 ∈ 𝒫 ( Vtx ‘ 𝐺 ) ↔ 𝑒 ⊆ 𝑉 ) |
19 |
|
sstr |
⊢ ( ( { 𝑁 , 𝑛 } ⊆ 𝑒 ∧ 𝑒 ⊆ 𝑉 ) → { 𝑁 , 𝑛 } ⊆ 𝑉 ) |
20 |
|
prssg |
⊢ ( ( 𝑁 ∈ 𝑋 ∧ 𝑛 ∈ V ) → ( ( 𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉 ) ↔ { 𝑁 , 𝑛 } ⊆ 𝑉 ) ) |
21 |
20
|
bicomd |
⊢ ( ( 𝑁 ∈ 𝑋 ∧ 𝑛 ∈ V ) → ( { 𝑁 , 𝑛 } ⊆ 𝑉 ↔ ( 𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉 ) ) ) |
22 |
21
|
elvd |
⊢ ( 𝑁 ∈ 𝑋 → ( { 𝑁 , 𝑛 } ⊆ 𝑉 ↔ ( 𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉 ) ) ) |
23 |
|
simpl |
⊢ ( ( 𝑁 ∈ 𝑉 ∧ 𝑛 ∈ 𝑉 ) → 𝑁 ∈ 𝑉 ) |
24 |
22 23
|
syl6bi |
⊢ ( 𝑁 ∈ 𝑋 → ( { 𝑁 , 𝑛 } ⊆ 𝑉 → 𝑁 ∈ 𝑉 ) ) |
25 |
19 24
|
syl5com |
⊢ ( ( { 𝑁 , 𝑛 } ⊆ 𝑒 ∧ 𝑒 ⊆ 𝑉 ) → ( 𝑁 ∈ 𝑋 → 𝑁 ∈ 𝑉 ) ) |
26 |
25
|
ex |
⊢ ( { 𝑁 , 𝑛 } ⊆ 𝑒 → ( 𝑒 ⊆ 𝑉 → ( 𝑁 ∈ 𝑋 → 𝑁 ∈ 𝑉 ) ) ) |
27 |
26
|
com13 |
⊢ ( 𝑁 ∈ 𝑋 → ( 𝑒 ⊆ 𝑉 → ( { 𝑁 , 𝑛 } ⊆ 𝑒 → 𝑁 ∈ 𝑉 ) ) ) |
28 |
27
|
ad3antlr |
⊢ ( ( ( ( 𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋 ) ∧ 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ) ∧ 𝑒 ∈ 𝐸 ) → ( 𝑒 ⊆ 𝑉 → ( { 𝑁 , 𝑛 } ⊆ 𝑒 → 𝑁 ∈ 𝑉 ) ) ) |
29 |
18 28
|
syl5bi |
⊢ ( ( ( ( 𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋 ) ∧ 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ) ∧ 𝑒 ∈ 𝐸 ) → ( 𝑒 ∈ 𝒫 ( Vtx ‘ 𝐺 ) → ( { 𝑁 , 𝑛 } ⊆ 𝑒 → 𝑁 ∈ 𝑉 ) ) ) |
30 |
14 29
|
mpd |
⊢ ( ( ( ( 𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋 ) ∧ 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ) ∧ 𝑒 ∈ 𝐸 ) → ( { 𝑁 , 𝑛 } ⊆ 𝑒 → 𝑁 ∈ 𝑉 ) ) |
31 |
30
|
rexlimdva |
⊢ ( ( ( 𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋 ) ∧ 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ) → ( ∃ 𝑒 ∈ 𝐸 { 𝑁 , 𝑛 } ⊆ 𝑒 → 𝑁 ∈ 𝑉 ) ) |
32 |
31
|
con3rr3 |
⊢ ( ¬ 𝑁 ∈ 𝑉 → ( ( ( 𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋 ) ∧ 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ) → ¬ ∃ 𝑒 ∈ 𝐸 { 𝑁 , 𝑛 } ⊆ 𝑒 ) ) |
33 |
32
|
expdimp |
⊢ ( ( ¬ 𝑁 ∈ 𝑉 ∧ ( 𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋 ) ) → ( 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) → ¬ ∃ 𝑒 ∈ 𝐸 { 𝑁 , 𝑛 } ⊆ 𝑒 ) ) |
34 |
33
|
ralrimiv |
⊢ ( ( ¬ 𝑁 ∈ 𝑉 ∧ ( 𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋 ) ) → ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ¬ ∃ 𝑒 ∈ 𝐸 { 𝑁 , 𝑛 } ⊆ 𝑒 ) |
35 |
|
rabeq0 |
⊢ ( { 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ∣ ∃ 𝑒 ∈ 𝐸 { 𝑁 , 𝑛 } ⊆ 𝑒 } = ∅ ↔ ∀ 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ¬ ∃ 𝑒 ∈ 𝐸 { 𝑁 , 𝑛 } ⊆ 𝑒 ) |
36 |
34 35
|
sylibr |
⊢ ( ( ¬ 𝑁 ∈ 𝑉 ∧ ( 𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋 ) ) → { 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ∣ ∃ 𝑒 ∈ 𝐸 { 𝑁 , 𝑛 } ⊆ 𝑒 } = ∅ ) |
37 |
8 36
|
eqtr4d |
⊢ ( ( ¬ 𝑁 ∈ 𝑉 ∧ ( 𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋 ) ) → ( 𝐺 NeighbVtx 𝑁 ) = { 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ∣ ∃ 𝑒 ∈ 𝐸 { 𝑁 , 𝑛 } ⊆ 𝑒 } ) |
38 |
37
|
ex |
⊢ ( ¬ 𝑁 ∈ 𝑉 → ( ( 𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋 ) → ( 𝐺 NeighbVtx 𝑁 ) = { 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ∣ ∃ 𝑒 ∈ 𝐸 { 𝑁 , 𝑛 } ⊆ 𝑒 } ) ) |
39 |
4 38
|
pm2.61i |
⊢ ( ( 𝐺 ∈ UHGraph ∧ 𝑁 ∈ 𝑋 ) → ( 𝐺 NeighbVtx 𝑁 ) = { 𝑛 ∈ ( 𝑉 ∖ { 𝑁 } ) ∣ ∃ 𝑒 ∈ 𝐸 { 𝑁 , 𝑛 } ⊆ 𝑒 } ) |