Metamath Proof Explorer
Description: In dimension zero, any two points are equal. (Contributed by Thierry
Arnoux, 12-Apr-2019)
|
|
Ref |
Expression |
|
Hypotheses |
tgbtwndiff.p |
⊢ 𝑃 = ( Base ‘ 𝐺 ) |
|
|
tgbtwndiff.d |
⊢ − = ( dist ‘ 𝐺 ) |
|
|
tgbtwndiff.i |
⊢ 𝐼 = ( Itv ‘ 𝐺 ) |
|
|
tgbtwndiff.g |
⊢ ( 𝜑 → 𝐺 ∈ TarskiG ) |
|
|
tgbtwndiff.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
|
|
tgbtwndiff.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) |
|
|
tgldim0itv.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑃 ) |
|
|
tgldim0itv.p |
⊢ ( 𝜑 → ( ♯ ‘ 𝑃 ) = 1 ) |
|
Assertion |
tgldim0itv |
⊢ ( 𝜑 → 𝐴 ∈ ( 𝐵 𝐼 𝐶 ) ) |
Proof
Step |
Hyp |
Ref |
Expression |
1 |
|
tgbtwndiff.p |
⊢ 𝑃 = ( Base ‘ 𝐺 ) |
2 |
|
tgbtwndiff.d |
⊢ − = ( dist ‘ 𝐺 ) |
3 |
|
tgbtwndiff.i |
⊢ 𝐼 = ( Itv ‘ 𝐺 ) |
4 |
|
tgbtwndiff.g |
⊢ ( 𝜑 → 𝐺 ∈ TarskiG ) |
5 |
|
tgbtwndiff.a |
⊢ ( 𝜑 → 𝐴 ∈ 𝑃 ) |
6 |
|
tgbtwndiff.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝑃 ) |
7 |
|
tgldim0itv.c |
⊢ ( 𝜑 → 𝐶 ∈ 𝑃 ) |
8 |
|
tgldim0itv.p |
⊢ ( 𝜑 → ( ♯ ‘ 𝑃 ) = 1 ) |
9 |
1 8 5 6
|
tgldim0eq |
⊢ ( 𝜑 → 𝐴 = 𝐵 ) |
10 |
1 2 3 4 6 7
|
tgbtwntriv1 |
⊢ ( 𝜑 → 𝐵 ∈ ( 𝐵 𝐼 𝐶 ) ) |
11 |
9 10
|
eqeltrd |
⊢ ( 𝜑 → 𝐴 ∈ ( 𝐵 𝐼 𝐶 ) ) |