Step |
Hyp |
Ref |
Expression |
1 |
|
simp1 |
⊢ ( ( 𝐴 ⊆ ℤ ∧ ∃ 𝑥 ∈ ℤ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ∧ 𝐵 ∈ 𝐴 ) → 𝐴 ⊆ ℤ ) |
2 |
|
zssre |
⊢ ℤ ⊆ ℝ |
3 |
1 2
|
sstrdi |
⊢ ( ( 𝐴 ⊆ ℤ ∧ ∃ 𝑥 ∈ ℤ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ∧ 𝐵 ∈ 𝐴 ) → 𝐴 ⊆ ℝ ) |
4 |
|
simp3 |
⊢ ( ( 𝐴 ⊆ ℤ ∧ ∃ 𝑥 ∈ ℤ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ∧ 𝐵 ∈ 𝐴 ) → 𝐵 ∈ 𝐴 ) |
5 |
3 4
|
sseldd |
⊢ ( ( 𝐴 ⊆ ℤ ∧ ∃ 𝑥 ∈ ℤ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ∧ 𝐵 ∈ 𝐴 ) → 𝐵 ∈ ℝ ) |
6 |
4
|
ne0d |
⊢ ( ( 𝐴 ⊆ ℤ ∧ ∃ 𝑥 ∈ ℤ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ∧ 𝐵 ∈ 𝐴 ) → 𝐴 ≠ ∅ ) |
7 |
|
simp2 |
⊢ ( ( 𝐴 ⊆ ℤ ∧ ∃ 𝑥 ∈ ℤ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ∧ 𝐵 ∈ 𝐴 ) → ∃ 𝑥 ∈ ℤ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) |
8 |
|
suprzcl2 |
⊢ ( ( 𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℤ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) → sup ( 𝐴 , ℝ , < ) ∈ 𝐴 ) |
9 |
1 6 7 8
|
syl3anc |
⊢ ( ( 𝐴 ⊆ ℤ ∧ ∃ 𝑥 ∈ ℤ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ∧ 𝐵 ∈ 𝐴 ) → sup ( 𝐴 , ℝ , < ) ∈ 𝐴 ) |
10 |
3 9
|
sseldd |
⊢ ( ( 𝐴 ⊆ ℤ ∧ ∃ 𝑥 ∈ ℤ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ∧ 𝐵 ∈ 𝐴 ) → sup ( 𝐴 , ℝ , < ) ∈ ℝ ) |
11 |
|
ltso |
⊢ < Or ℝ |
12 |
11
|
a1i |
⊢ ( ( 𝐴 ⊆ ℤ ∧ ∃ 𝑥 ∈ ℤ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ∧ 𝐵 ∈ 𝐴 ) → < Or ℝ ) |
13 |
|
zsupss |
⊢ ( ( 𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℤ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑥 → ∃ 𝑧 ∈ 𝐴 𝑦 < 𝑧 ) ) ) |
14 |
1 6 7 13
|
syl3anc |
⊢ ( ( 𝐴 ⊆ ℤ ∧ ∃ 𝑥 ∈ ℤ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ∧ 𝐵 ∈ 𝐴 ) → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑥 → ∃ 𝑧 ∈ 𝐴 𝑦 < 𝑧 ) ) ) |
15 |
|
ssrexv |
⊢ ( 𝐴 ⊆ ℝ → ( ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑥 → ∃ 𝑧 ∈ 𝐴 𝑦 < 𝑧 ) ) → ∃ 𝑥 ∈ ℝ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑥 → ∃ 𝑧 ∈ 𝐴 𝑦 < 𝑧 ) ) ) ) |
16 |
3 14 15
|
sylc |
⊢ ( ( 𝐴 ⊆ ℤ ∧ ∃ 𝑥 ∈ ℤ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ∧ 𝐵 ∈ 𝐴 ) → ∃ 𝑥 ∈ ℝ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑥 → ∃ 𝑧 ∈ 𝐴 𝑦 < 𝑧 ) ) ) |
17 |
12 16
|
supub |
⊢ ( ( 𝐴 ⊆ ℤ ∧ ∃ 𝑥 ∈ ℤ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ∧ 𝐵 ∈ 𝐴 ) → ( 𝐵 ∈ 𝐴 → ¬ sup ( 𝐴 , ℝ , < ) < 𝐵 ) ) |
18 |
4 17
|
mpd |
⊢ ( ( 𝐴 ⊆ ℤ ∧ ∃ 𝑥 ∈ ℤ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ∧ 𝐵 ∈ 𝐴 ) → ¬ sup ( 𝐴 , ℝ , < ) < 𝐵 ) |
19 |
5 10 18
|
nltled |
⊢ ( ( 𝐴 ⊆ ℤ ∧ ∃ 𝑥 ∈ ℤ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ∧ 𝐵 ∈ 𝐴 ) → 𝐵 ≤ sup ( 𝐴 , ℝ , < ) ) |