Step |
Hyp |
Ref |
Expression |
1 |
|
infxrge0glb.a |
⊢ ( 𝜑 → 𝐴 ⊆ ( 0 [,] +∞ ) ) |
2 |
|
infxrge0glb.b |
⊢ ( 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 |
⊢ ( 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 1
|
infglbb |
⊢ ( ( 𝜑 ∧ 𝐵 ∈ ( 0 [,] +∞ ) ) → ( inf ( 𝐴 , ( 0 [,] +∞ ) , < ) < 𝐵 ↔ ∃ 𝑧 ∈ 𝐴 𝑧 < 𝐵 ) ) |
11 |
2 10
|
mpdan |
⊢ ( 𝜑 → ( inf ( 𝐴 , ( 0 [,] +∞ ) , < ) < 𝐵 ↔ ∃ 𝑧 ∈ 𝐴 𝑧 < 𝐵 ) ) |
12 |
|
breq1 |
⊢ ( 𝑥 = 𝑧 → ( 𝑥 < 𝐵 ↔ 𝑧 < 𝐵 ) ) |
13 |
12
|
cbvrexvw |
⊢ ( ∃ 𝑥 ∈ 𝐴 𝑥 < 𝐵 ↔ ∃ 𝑧 ∈ 𝐴 𝑧 < 𝐵 ) |
14 |
11 13
|
bitr4di |
⊢ ( 𝜑 → ( inf ( 𝐴 , ( 0 [,] +∞ ) , < ) < 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 𝑥 < 𝐵 ) ) |