Step |
Hyp |
Ref |
Expression |
1 |
|
simpr |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ 𝐵 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑥 ) → ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑥 ) |
2 |
|
simpl1 |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ 𝐵 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑥 ) → 𝐴 ⊆ ℝ ) |
3 |
|
simpl2 |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ 𝐵 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑥 ) → 𝐴 ≠ ∅ ) |
4 |
|
breq1 |
⊢ ( 𝑧 = 𝐵 → ( 𝑧 ≤ 𝑥 ↔ 𝐵 ≤ 𝑥 ) ) |
5 |
4
|
ralbidv |
⊢ ( 𝑧 = 𝐵 → ( ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝑥 ↔ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑥 ) ) |
6 |
5
|
rspcev |
⊢ ( ( 𝐵 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑥 ) → ∃ 𝑧 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝑥 ) |
7 |
6
|
3ad2antl3 |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ 𝐵 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑥 ) → ∃ 𝑧 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝑥 ) |
8 |
|
simpl3 |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ 𝐵 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑥 ) → 𝐵 ∈ ℝ ) |
9 |
|
infregelb |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑧 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝑧 ≤ 𝑥 ) ∧ 𝐵 ∈ ℝ ) → ( 𝐵 ≤ inf ( 𝐴 , ℝ , < ) ↔ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑥 ) ) |
10 |
2 3 7 8 9
|
syl31anc |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ 𝐵 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑥 ) → ( 𝐵 ≤ inf ( 𝐴 , ℝ , < ) ↔ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑥 ) ) |
11 |
1 10
|
mpbird |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ 𝐵 ∈ ℝ ) ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑥 ) → 𝐵 ≤ inf ( 𝐴 , ℝ , < ) ) |