Step |
Hyp |
Ref |
Expression |
1 |
|
infxrge0lb.a |
⊢ ( 𝜑 → 𝐴 ⊆ ( 0 [,] +∞ ) ) |
2 |
|
infxrge0lb.b |
⊢ ( 𝜑 → 𝐵 ∈ 𝐴 ) |
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 |
⊢ ( 𝐴 ⊆ ( 0 [,] +∞ ) → ∃ 𝑥 ∈ ( 0 [,] +∞ ) ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑥 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) |
9 |
1 8
|
syl |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ( 0 [,] +∞ ) ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀ 𝑦 ∈ ( 0 [,] +∞ ) ( 𝑥 < 𝑦 → ∃ 𝑧 ∈ 𝐴 𝑧 < 𝑦 ) ) ) |
10 |
7 9
|
infcl |
⊢ ( 𝜑 → inf ( 𝐴 , ( 0 [,] +∞ ) , < ) ∈ ( 0 [,] +∞ ) ) |
11 |
3 10
|
sselid |
⊢ ( 𝜑 → inf ( 𝐴 , ( 0 [,] +∞ ) , < ) ∈ ℝ* ) |
12 |
1 2
|
sseldd |
⊢ ( 𝜑 → 𝐵 ∈ ( 0 [,] +∞ ) ) |
13 |
3 12
|
sselid |
⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) |
14 |
7 9
|
inflb |
⊢ ( 𝜑 → ( 𝐵 ∈ 𝐴 → ¬ 𝐵 < inf ( 𝐴 , ( 0 [,] +∞ ) , < ) ) ) |
15 |
2 14
|
mpd |
⊢ ( 𝜑 → ¬ 𝐵 < inf ( 𝐴 , ( 0 [,] +∞ ) , < ) ) |
16 |
11 13 15
|
xrnltled |
⊢ ( 𝜑 → inf ( 𝐴 , ( 0 [,] +∞ ) , < ) ≤ 𝐵 ) |