Step |
Hyp |
Ref |
Expression |
1 |
|
tgbtwnconn.p |
⊢ 𝑃 = ( Base ‘ 𝐺 ) |
2 |
|
tgbtwnconn.i |
⊢ 𝐼 = ( Itv ‘ 𝐺 ) |
3 |
|
tgbtwnconn.g |
⊢ ( 𝜑 → 𝐺 ∈ TarskiG ) |
4 |
|
tgbtwnconn.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
5 |
|
tgbtwnconn.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) |
6 |
|
tgbtwnconn.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑃 ) |
7 |
|
tgbtwnconn.d |
⊢ ( 𝜑 → 𝐷 ∈ 𝑃 ) |
8 |
|
tgbtwnconnln1.l |
⊢ 𝐿 = ( LineG ‘ 𝐺 ) |
9 |
|
tgbtwnconnln1.1 |
⊢ ( 𝜑 → 𝐴 ≠ 𝐵 ) |
10 |
|
tgbtwnconnln1.2 |
⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) ) |
11 |
|
tgbtwnconnln1.3 |
⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐷 ) ) |
12 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) ) → 𝐺 ∈ TarskiG ) |
13 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) ) → 𝐶 ∈ 𝑃 ) |
14 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) ) → 𝐷 ∈ 𝑃 ) |
15 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) ) → 𝐵 ∈ 𝑃 ) |
16 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) ) → 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) ) |
17 |
1 8 2 12 13 14 15 16
|
btwncolg2 |
⊢ ( ( 𝜑 ∧ 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) ) → ( 𝐵 ∈ ( 𝐶 𝐿 𝐷 ) ∨ 𝐶 = 𝐷 ) ) |
18 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐵 𝐼 𝐶 ) ) → 𝐺 ∈ TarskiG ) |
19 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐵 𝐼 𝐶 ) ) → 𝐶 ∈ 𝑃 ) |
20 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐵 𝐼 𝐶 ) ) → 𝐷 ∈ 𝑃 ) |
21 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐵 𝐼 𝐶 ) ) → 𝐵 ∈ 𝑃 ) |
22 |
|
eqid |
⊢ ( dist ‘ 𝐺 ) = ( dist ‘ 𝐺 ) |
23 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐵 𝐼 𝐶 ) ) → 𝐷 ∈ ( 𝐵 𝐼 𝐶 ) ) |
24 |
1 22 2 18 21 20 19 23
|
tgbtwncom |
⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐵 𝐼 𝐶 ) ) → 𝐷 ∈ ( 𝐶 𝐼 𝐵 ) ) |
25 |
1 8 2 18 19 20 21 24
|
btwncolg3 |
⊢ ( ( 𝜑 ∧ 𝐷 ∈ ( 𝐵 𝐼 𝐶 ) ) → ( 𝐵 ∈ ( 𝐶 𝐿 𝐷 ) ∨ 𝐶 = 𝐷 ) ) |
26 |
1 2 3 4 5 6 7 9 10 11
|
tgbtwnconn2 |
⊢ ( 𝜑 → ( 𝐶 ∈ ( 𝐵 𝐼 𝐷 ) ∨ 𝐷 ∈ ( 𝐵 𝐼 𝐶 ) ) ) |
27 |
17 25 26
|
mpjaodan |
⊢ ( 𝜑 → ( 𝐵 ∈ ( 𝐶 𝐿 𝐷 ) ∨ 𝐶 = 𝐷 ) ) |