Step |
Hyp |
Ref |
Expression |
1 |
|
hpg.p |
⊢ 𝑃 = ( Base ‘ 𝐺 ) |
2 |
|
hpg.d |
⊢ − = ( dist ‘ 𝐺 ) |
3 |
|
hpg.i |
⊢ 𝐼 = ( Itv ‘ 𝐺 ) |
4 |
|
hpg.o |
⊢ 𝑂 = { 〈 𝑎 , 𝑏 〉 ∣ ( ( 𝑎 ∈ ( 𝑃 ∖ 𝐷 ) ∧ 𝑏 ∈ ( 𝑃 ∖ 𝐷 ) ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝑎 𝐼 𝑏 ) ) } |
5 |
|
islnoppd.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
6 |
|
islnoppd.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) |
7 |
|
islnoppd.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝐷 ) |
8 |
|
islnoppd.1 |
⊢ ( 𝜑 → ¬ 𝐴 ∈ 𝐷 ) |
9 |
|
islnoppd.2 |
⊢ ( 𝜑 → ¬ 𝐵 ∈ 𝐷 ) |
10 |
|
islnoppd.3 |
⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ) |
11 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑡 = 𝐶 ) → 𝑡 = 𝐶 ) |
12 |
11
|
eleq1d |
⊢ ( ( 𝜑 ∧ 𝑡 = 𝐶 ) → ( 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ↔ 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ) ) |
13 |
7 12 10
|
rspcedvd |
⊢ ( 𝜑 → ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ) |
14 |
8 9 13
|
jca31 |
⊢ ( 𝜑 → ( ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ) ) |
15 |
1 2 3 4 5 6
|
islnopp |
⊢ ( 𝜑 → ( 𝐴 𝑂 𝐵 ↔ ( ( ¬ 𝐴 ∈ 𝐷 ∧ ¬ 𝐵 ∈ 𝐷 ) ∧ ∃ 𝑡 ∈ 𝐷 𝑡 ∈ ( 𝐴 𝐼 𝐵 ) ) ) ) |
16 |
14 15
|
mpbird |
⊢ ( 𝜑 → 𝐴 𝑂 𝐵 ) |