Step |
Hyp |
Ref |
Expression |
1 |
|
xrge0infssd.1 |
⊢ ( 𝜑 → 𝐶 ⊆ 𝐵 ) |
2 |
|
xrge0infssd.2 |
⊢ ( 𝜑 → 𝐵 ⊆ ( 0 [,] +∞ ) ) |
3 |
|
iccssxr |
⊢ ( 0 [,] +∞ ) ⊆ ℝ* |
4 |
|
xrltso |
⊢ < Or ℝ* |
5 |
|
soss |
⊢ ( ( 0 [,] +∞ ) ⊆ ℝ* → ( < Or ℝ* → < Or ( 0 [,] +∞ ) ) ) |
6 |
3 4 5
|
mp2 |
⊢ < Or ( 0 [,] +∞ ) |
7 |
6
|
a1i |
⊢ ( 𝜑 → < Or ( 0 [,] +∞ ) ) |
8 |
|
xrge0infss |
⊢ ( 𝐵 ⊆ ( 0 [,] +∞ ) → ∃ 𝑥 ∈ ( 0 [,] +∞ ) ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 < 𝑥 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑥 < 𝑦 → ∃ 𝑧 ∈ 𝐵 𝑧 < 𝑦 ) ) ) |
9 |
2 8
|
syl |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ( 0 [,] +∞ ) ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 < 𝑥 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑥 < 𝑦 → ∃ 𝑧 ∈ 𝐵 𝑧 < 𝑦 ) ) ) |
10 |
7 9
|
infcl |
⊢ ( 𝜑 → inf ( 𝐵 , ( 0 [,] +∞ ) , < ) ∈ ( 0 [,] +∞ ) ) |
11 |
3 10
|
sselid |
⊢ ( 𝜑 → inf ( 𝐵 , ( 0 [,] +∞ ) , < ) ∈ ℝ* ) |
12 |
1 2
|
sstrd |
⊢ ( 𝜑 → 𝐶 ⊆ ( 0 [,] +∞ ) ) |
13 |
|
xrge0infss |
⊢ ( 𝐶 ⊆ ( 0 [,] +∞ ) → ∃ 𝑥 ∈ ( 0 [,] +∞ ) ( ∀ 𝑦 ∈ 𝐶 ¬ 𝑦 < 𝑥 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑥 < 𝑦 → ∃ 𝑧 ∈ 𝐶 𝑧 < 𝑦 ) ) ) |
14 |
12 13
|
syl |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ( 0 [,] +∞ ) ( ∀ 𝑦 ∈ 𝐶 ¬ 𝑦 < 𝑥 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑥 < 𝑦 → ∃ 𝑧 ∈ 𝐶 𝑧 < 𝑦 ) ) ) |
15 |
7 14
|
infcl |
⊢ ( 𝜑 → inf ( 𝐶 , ( 0 [,] +∞ ) , < ) ∈ ( 0 [,] +∞ ) ) |
16 |
3 15
|
sselid |
⊢ ( 𝜑 → inf ( 𝐶 , ( 0 [,] +∞ ) , < ) ∈ ℝ* ) |
17 |
7 1 14 9
|
infssd |
⊢ ( 𝜑 → ¬ inf ( 𝐶 , ( 0 [,] +∞ ) , < ) < inf ( 𝐵 , ( 0 [,] +∞ ) , < ) ) |
18 |
11 16 17
|
xrnltled |
⊢ ( 𝜑 → inf ( 𝐵 , ( 0 [,] +∞ ) , < ) ≤ inf ( 𝐶 , ( 0 [,] +∞ ) , < ) ) |