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 |
|
tgbtwnconn3.1 |
⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 𝐼 𝐷 ) ) |
9 |
|
tgbtwnconn3.2 |
⊢ ( 𝜑 → 𝐶 ∈ ( 𝐴 𝐼 𝐷 ) ) |
10 |
|
eqid |
⊢ ( dist ‘ 𝐺 ) = ( dist ‘ 𝐺 ) |
11 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( ♯ ‘ 𝑃 ) = 1 ) → 𝐺 ∈ TarskiG ) |
12 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ ( ♯ ‘ 𝑃 ) = 1 ) → 𝐵 ∈ 𝑃 ) |
13 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ ( ♯ ‘ 𝑃 ) = 1 ) → 𝐴 ∈ 𝑃 ) |
14 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ ( ♯ ‘ 𝑃 ) = 1 ) → 𝐶 ∈ 𝑃 ) |
15 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( ♯ ‘ 𝑃 ) = 1 ) → ( ♯ ‘ 𝑃 ) = 1 ) |
16 |
1 10 2 11 12 13 14 15
|
tgldim0itv |
⊢ ( ( 𝜑 ∧ ( ♯ ‘ 𝑃 ) = 1 ) → 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) ) |
17 |
16
|
orcd |
⊢ ( ( 𝜑 ∧ ( ♯ ‘ 𝑃 ) = 1 ) → ( 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) ∨ 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ) ) |
18 |
3
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑝 ) ∧ 𝐴 ≠ 𝑝 ) ) → 𝐺 ∈ TarskiG ) |
19 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑝 ) ∧ 𝐴 ≠ 𝑝 ) ) → 𝑝 ∈ 𝑃 ) |
20 |
4
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑝 ) ∧ 𝐴 ≠ 𝑝 ) ) → 𝐴 ∈ 𝑃 ) |
21 |
5
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑝 ) ∧ 𝐴 ≠ 𝑝 ) ) → 𝐵 ∈ 𝑃 ) |
22 |
6
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑝 ) ∧ 𝐴 ≠ 𝑝 ) ) → 𝐶 ∈ 𝑃 ) |
23 |
|
simprr |
⊢ ( ( ( ( 𝜑 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑝 ) ∧ 𝐴 ≠ 𝑝 ) ) → 𝐴 ≠ 𝑝 ) |
24 |
23
|
necomd |
⊢ ( ( ( ( 𝜑 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑝 ) ∧ 𝐴 ≠ 𝑝 ) ) → 𝑝 ≠ 𝐴 ) |
25 |
7
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑝 ) ∧ 𝐴 ≠ 𝑝 ) ) → 𝐷 ∈ 𝑃 ) |
26 |
8
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑝 ) ∧ 𝐴 ≠ 𝑝 ) ) → 𝐵 ∈ ( 𝐴 𝐼 𝐷 ) ) |
27 |
|
simprl |
⊢ ( ( ( ( 𝜑 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑝 ) ∧ 𝐴 ≠ 𝑝 ) ) → 𝐴 ∈ ( 𝐷 𝐼 𝑝 ) ) |
28 |
1 10 2 18 25 20 19 27
|
tgbtwncom |
⊢ ( ( ( ( 𝜑 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑝 ) ∧ 𝐴 ≠ 𝑝 ) ) → 𝐴 ∈ ( 𝑝 𝐼 𝐷 ) ) |
29 |
1 10 2 18 21 20 19 25 26 28
|
tgbtwnintr |
⊢ ( ( ( ( 𝜑 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑝 ) ∧ 𝐴 ≠ 𝑝 ) ) → 𝐴 ∈ ( 𝐵 𝐼 𝑝 ) ) |
30 |
1 10 2 18 21 20 19 29
|
tgbtwncom |
⊢ ( ( ( ( 𝜑 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑝 ) ∧ 𝐴 ≠ 𝑝 ) ) → 𝐴 ∈ ( 𝑝 𝐼 𝐵 ) ) |
31 |
9
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑝 ) ∧ 𝐴 ≠ 𝑝 ) ) → 𝐶 ∈ ( 𝐴 𝐼 𝐷 ) ) |
32 |
1 10 2 18 20 22 25 31
|
tgbtwncom |
⊢ ( ( ( ( 𝜑 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑝 ) ∧ 𝐴 ≠ 𝑝 ) ) → 𝐶 ∈ ( 𝐷 𝐼 𝐴 ) ) |
33 |
1 10 2 18 25 22 20 19 32 27
|
tgbtwnexch3 |
⊢ ( ( ( ( 𝜑 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑝 ) ∧ 𝐴 ≠ 𝑝 ) ) → 𝐴 ∈ ( 𝐶 𝐼 𝑝 ) ) |
34 |
1 10 2 18 22 20 19 33
|
tgbtwncom |
⊢ ( ( ( ( 𝜑 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑝 ) ∧ 𝐴 ≠ 𝑝 ) ) → 𝐴 ∈ ( 𝑝 𝐼 𝐶 ) ) |
35 |
1 2 18 19 20 21 22 24 30 34
|
tgbtwnconn2 |
⊢ ( ( ( ( 𝜑 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) ∧ 𝑝 ∈ 𝑃 ) ∧ ( 𝐴 ∈ ( 𝐷 𝐼 𝑝 ) ∧ 𝐴 ≠ 𝑝 ) ) → ( 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) ∨ 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ) ) |
36 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → 𝐺 ∈ TarskiG ) |
37 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → 𝐷 ∈ 𝑃 ) |
38 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → 𝐴 ∈ 𝑃 ) |
39 |
|
simpr |
⊢ ( ( 𝜑 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → 2 ≤ ( ♯ ‘ 𝑃 ) ) |
40 |
1 10 2 36 37 38 39
|
tgbtwndiff |
⊢ ( ( 𝜑 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → ∃ 𝑝 ∈ 𝑃 ( 𝐴 ∈ ( 𝐷 𝐼 𝑝 ) ∧ 𝐴 ≠ 𝑝 ) ) |
41 |
35 40
|
r19.29a |
⊢ ( ( 𝜑 ∧ 2 ≤ ( ♯ ‘ 𝑃 ) ) → ( 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) ∨ 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ) ) |
42 |
1 4
|
tgldimor |
⊢ ( 𝜑 → ( ( ♯ ‘ 𝑃 ) = 1 ∨ 2 ≤ ( ♯ ‘ 𝑃 ) ) ) |
43 |
17 41 42
|
mpjaodan |
⊢ ( 𝜑 → ( 𝐵 ∈ ( 𝐴 𝐼 𝐶 ) ∨ 𝐶 ∈ ( 𝐴 𝐼 𝐵 ) ) ) |