Step |
Hyp |
Ref |
Expression |
1 |
|
supfirege.1 |
⊢ ( 𝜑 → 𝐵 ⊆ ℝ ) |
2 |
|
supfirege.2 |
⊢ ( 𝜑 → 𝐵 ∈ Fin ) |
3 |
|
supfirege.3 |
⊢ ( 𝜑 → 𝐶 ∈ 𝐵 ) |
4 |
|
supfirege.4 |
⊢ ( 𝜑 → 𝑆 = sup ( 𝐵 , ℝ , < ) ) |
5 |
|
ltso |
⊢ < Or ℝ |
6 |
5
|
a1i |
⊢ ( 𝜑 → < Or ℝ ) |
7 |
6 1 2 3 4
|
supgtoreq |
⊢ ( 𝜑 → ( 𝐶 < 𝑆 ∨ 𝐶 = 𝑆 ) ) |
8 |
1 3
|
sseldd |
⊢ ( 𝜑 → 𝐶 ∈ ℝ ) |
9 |
3
|
ne0d |
⊢ ( 𝜑 → 𝐵 ≠ ∅ ) |
10 |
|
fisupcl |
⊢ ( ( < Or ℝ ∧ ( 𝐵 ∈ Fin ∧ 𝐵 ≠ ∅ ∧ 𝐵 ⊆ ℝ ) ) → sup ( 𝐵 , ℝ , < ) ∈ 𝐵 ) |
11 |
6 2 9 1 10
|
syl13anc |
⊢ ( 𝜑 → sup ( 𝐵 , ℝ , < ) ∈ 𝐵 ) |
12 |
4 11
|
eqeltrd |
⊢ ( 𝜑 → 𝑆 ∈ 𝐵 ) |
13 |
1 12
|
sseldd |
⊢ ( 𝜑 → 𝑆 ∈ ℝ ) |
14 |
8 13
|
leloed |
⊢ ( 𝜑 → ( 𝐶 ≤ 𝑆 ↔ ( 𝐶 < 𝑆 ∨ 𝐶 = 𝑆 ) ) ) |
15 |
7 14
|
mpbird |
⊢ ( 𝜑 → 𝐶 ≤ 𝑆 ) |