Step |
Hyp |
Ref |
Expression |
1 |
|
ssel2 |
⊢ ( ( 𝐴 ⊆ ℝ* ∧ 𝐵 ∈ 𝐴 ) → 𝐵 ∈ ℝ* ) |
2 |
|
supxrcl |
⊢ ( 𝐴 ⊆ ℝ* → sup ( 𝐴 , ℝ* , < ) ∈ ℝ* ) |
3 |
2
|
adantr |
⊢ ( ( 𝐴 ⊆ ℝ* ∧ 𝐵 ∈ 𝐴 ) → sup ( 𝐴 , ℝ* , < ) ∈ ℝ* ) |
4 |
|
xrltso |
⊢ < Or ℝ* |
5 |
4
|
a1i |
⊢ ( 𝐴 ⊆ ℝ* → < Or ℝ* ) |
6 |
|
xrsupss |
⊢ ( 𝐴 ⊆ ℝ* → ∃ 𝑥 ∈ ℝ* ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ* ( 𝑦 < 𝑥 → ∃ 𝑧 ∈ 𝐴 𝑦 < 𝑧 ) ) ) |
7 |
5 6
|
supub |
⊢ ( 𝐴 ⊆ ℝ* → ( 𝐵 ∈ 𝐴 → ¬ sup ( 𝐴 , ℝ* , < ) < 𝐵 ) ) |
8 |
7
|
imp |
⊢ ( ( 𝐴 ⊆ ℝ* ∧ 𝐵 ∈ 𝐴 ) → ¬ sup ( 𝐴 , ℝ* , < ) < 𝐵 ) |
9 |
1 3 8
|
xrnltled |
⊢ ( ( 𝐴 ⊆ ℝ* ∧ 𝐵 ∈ 𝐴 ) → 𝐵 ≤ sup ( 𝐴 , ℝ* , < ) ) |