Step |
Hyp |
Ref |
Expression |
1 |
|
btwnlng1.p |
⊢ 𝑃 = ( Base ‘ 𝐺 ) |
2 |
|
btwnlng1.i |
⊢ 𝐼 = ( Itv ‘ 𝐺 ) |
3 |
|
btwnlng1.l |
⊢ 𝐿 = ( LineG ‘ 𝐺 ) |
4 |
|
btwnlng1.g |
⊢ ( 𝜑 → 𝐺 ∈ TarskiG ) |
5 |
|
btwnlng1.x |
⊢ ( 𝜑 → 𝑋 ∈ 𝑃 ) |
6 |
|
btwnlng1.y |
⊢ ( 𝜑 → 𝑌 ∈ 𝑃 ) |
7 |
|
btwnlng1.z |
⊢ ( 𝜑 → 𝑍 ∈ 𝑃 ) |
8 |
|
btwnlng1.d |
⊢ ( 𝜑 → 𝑋 ≠ 𝑌 ) |
9 |
|
lnrot1.1 |
⊢ ( 𝜑 → 𝑌 ∈ ( 𝑍 𝐿 𝑋 ) ) |
10 |
|
lnrot1.2 |
⊢ ( 𝜑 → 𝑍 ≠ 𝑋 ) |
11 |
|
eqid |
⊢ ( dist ‘ 𝐺 ) = ( dist ‘ 𝐺 ) |
12 |
1 11 2 4 6 7 5
|
tgbtwncomb |
⊢ ( 𝜑 → ( 𝑍 ∈ ( 𝑌 𝐼 𝑋 ) ↔ 𝑍 ∈ ( 𝑋 𝐼 𝑌 ) ) ) |
13 |
|
biidd |
⊢ ( 𝜑 → ( 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ↔ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ) ) |
14 |
1 11 2 4 7 6 5
|
tgbtwncomb |
⊢ ( 𝜑 → ( 𝑌 ∈ ( 𝑍 𝐼 𝑋 ) ↔ 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) ) ) |
15 |
12 13 14
|
3orbi123d |
⊢ ( 𝜑 → ( ( 𝑍 ∈ ( 𝑌 𝐼 𝑋 ) ∨ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑍 𝐼 𝑋 ) ) ↔ ( 𝑍 ∈ ( 𝑋 𝐼 𝑌 ) ∨ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) ) ) ) |
16 |
|
3orrot |
⊢ ( ( 𝑌 ∈ ( 𝑍 𝐼 𝑋 ) ∨ 𝑍 ∈ ( 𝑌 𝐼 𝑋 ) ∨ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ) ↔ ( 𝑍 ∈ ( 𝑌 𝐼 𝑋 ) ∨ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑍 𝐼 𝑋 ) ) ) |
17 |
16
|
a1i |
⊢ ( 𝜑 → ( ( 𝑌 ∈ ( 𝑍 𝐼 𝑋 ) ∨ 𝑍 ∈ ( 𝑌 𝐼 𝑋 ) ∨ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ) ↔ ( 𝑍 ∈ ( 𝑌 𝐼 𝑋 ) ∨ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑍 𝐼 𝑋 ) ) ) ) |
18 |
1 3 2 4 5 6 8 7
|
tgellng |
⊢ ( 𝜑 → ( 𝑍 ∈ ( 𝑋 𝐿 𝑌 ) ↔ ( 𝑍 ∈ ( 𝑋 𝐼 𝑌 ) ∨ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ∨ 𝑌 ∈ ( 𝑋 𝐼 𝑍 ) ) ) ) |
19 |
15 17 18
|
3bitr4rd |
⊢ ( 𝜑 → ( 𝑍 ∈ ( 𝑋 𝐿 𝑌 ) ↔ ( 𝑌 ∈ ( 𝑍 𝐼 𝑋 ) ∨ 𝑍 ∈ ( 𝑌 𝐼 𝑋 ) ∨ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ) ) ) |
20 |
1 3 2 4 7 5 10 6
|
tgellng |
⊢ ( 𝜑 → ( 𝑌 ∈ ( 𝑍 𝐿 𝑋 ) ↔ ( 𝑌 ∈ ( 𝑍 𝐼 𝑋 ) ∨ 𝑍 ∈ ( 𝑌 𝐼 𝑋 ) ∨ 𝑋 ∈ ( 𝑍 𝐼 𝑌 ) ) ) ) |
21 |
19 20
|
bitr4d |
⊢ ( 𝜑 → ( 𝑍 ∈ ( 𝑋 𝐿 𝑌 ) ↔ 𝑌 ∈ ( 𝑍 𝐿 𝑋 ) ) ) |
22 |
9 21
|
mpbird |
⊢ ( 𝜑 → 𝑍 ∈ ( 𝑋 𝐿 𝑌 ) ) |