Step |
Hyp |
Ref |
Expression |
1 |
|
zsupss |
⊢ ( ( 𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℤ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑥 → ∃ 𝑧 ∈ 𝐴 𝑦 < 𝑧 ) ) ) |
2 |
|
ssel2 |
⊢ ( ( 𝐴 ⊆ ℤ ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ ℤ ) |
3 |
2
|
zred |
⊢ ( ( 𝐴 ⊆ ℤ ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ ℝ ) |
4 |
|
ltso |
⊢ < Or ℝ |
5 |
4
|
a1i |
⊢ ( ⊤ → < Or ℝ ) |
6 |
5
|
eqsup |
⊢ ( ⊤ → ( ( 𝑥 ∈ ℝ ∧ ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑥 → ∃ 𝑧 ∈ 𝐴 𝑦 < 𝑧 ) ) → sup ( 𝐴 , ℝ , < ) = 𝑥 ) ) |
7 |
6
|
mptru |
⊢ ( ( 𝑥 ∈ ℝ ∧ ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑥 → ∃ 𝑧 ∈ 𝐴 𝑦 < 𝑧 ) ) → sup ( 𝐴 , ℝ , < ) = 𝑥 ) |
8 |
7
|
3expib |
⊢ ( 𝑥 ∈ ℝ → ( ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑥 → ∃ 𝑧 ∈ 𝐴 𝑦 < 𝑧 ) ) → sup ( 𝐴 , ℝ , < ) = 𝑥 ) ) |
9 |
3 8
|
syl |
⊢ ( ( 𝐴 ⊆ ℤ ∧ 𝑥 ∈ 𝐴 ) → ( ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑥 → ∃ 𝑧 ∈ 𝐴 𝑦 < 𝑧 ) ) → sup ( 𝐴 , ℝ , < ) = 𝑥 ) ) |
10 |
|
simpr |
⊢ ( ( 𝐴 ⊆ ℤ ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 ) |
11 |
|
eleq1 |
⊢ ( sup ( 𝐴 , ℝ , < ) = 𝑥 → ( sup ( 𝐴 , ℝ , < ) ∈ 𝐴 ↔ 𝑥 ∈ 𝐴 ) ) |
12 |
10 11
|
syl5ibrcom |
⊢ ( ( 𝐴 ⊆ ℤ ∧ 𝑥 ∈ 𝐴 ) → ( sup ( 𝐴 , ℝ , < ) = 𝑥 → sup ( 𝐴 , ℝ , < ) ∈ 𝐴 ) ) |
13 |
9 12
|
syld |
⊢ ( ( 𝐴 ⊆ ℤ ∧ 𝑥 ∈ 𝐴 ) → ( ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑥 → ∃ 𝑧 ∈ 𝐴 𝑦 < 𝑧 ) ) → sup ( 𝐴 , ℝ , < ) ∈ 𝐴 ) ) |
14 |
13
|
rexlimdva |
⊢ ( 𝐴 ⊆ ℤ → ( ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑥 → ∃ 𝑧 ∈ 𝐴 𝑦 < 𝑧 ) ) → sup ( 𝐴 , ℝ , < ) ∈ 𝐴 ) ) |
15 |
14
|
3ad2ant1 |
⊢ ( ( 𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℤ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) → ( ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑥 → ∃ 𝑧 ∈ 𝐴 𝑦 < 𝑧 ) ) → sup ( 𝐴 , ℝ , < ) ∈ 𝐴 ) ) |
16 |
1 15
|
mpd |
⊢ ( ( 𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℤ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) → sup ( 𝐴 , ℝ , < ) ∈ 𝐴 ) |