Step |
Hyp |
Ref |
Expression |
1 |
|
breq1 |
⊢ ( 𝑏 = 𝑦 → ( 𝑏 ≤ 𝑎 ↔ 𝑦 ≤ 𝑎 ) ) |
2 |
1
|
cbvralvw |
⊢ ( ∀ 𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 ↔ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑎 ) |
3 |
2
|
rexbii |
⊢ ( ∃ 𝑎 ∈ ℝ ∀ 𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 ↔ ∃ 𝑎 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑎 ) |
4 |
|
breq2 |
⊢ ( 𝑎 = 𝑥 → ( 𝑦 ≤ 𝑎 ↔ 𝑦 ≤ 𝑥 ) ) |
5 |
4
|
ralbidv |
⊢ ( 𝑎 = 𝑥 → ( ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑎 ↔ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) ) |
6 |
5
|
cbvrexvw |
⊢ ( ∃ 𝑎 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑎 ↔ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) |
7 |
3 6
|
bitri |
⊢ ( ∃ 𝑎 ∈ ℝ ∀ 𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 ↔ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ) |
8 |
|
renegcl |
⊢ ( 𝑎 ∈ ℝ → - 𝑎 ∈ ℝ ) |
9 |
|
elrabi |
⊢ ( 𝑦 ∈ { 𝑧 ∈ ℝ ∣ - 𝑧 ∈ 𝐴 } → 𝑦 ∈ ℝ ) |
10 |
|
negeq |
⊢ ( 𝑧 = 𝑦 → - 𝑧 = - 𝑦 ) |
11 |
10
|
eleq1d |
⊢ ( 𝑧 = 𝑦 → ( - 𝑧 ∈ 𝐴 ↔ - 𝑦 ∈ 𝐴 ) ) |
12 |
11
|
elrab3 |
⊢ ( 𝑦 ∈ ℝ → ( 𝑦 ∈ { 𝑧 ∈ ℝ ∣ - 𝑧 ∈ 𝐴 } ↔ - 𝑦 ∈ 𝐴 ) ) |
13 |
12
|
biimpd |
⊢ ( 𝑦 ∈ ℝ → ( 𝑦 ∈ { 𝑧 ∈ ℝ ∣ - 𝑧 ∈ 𝐴 } → - 𝑦 ∈ 𝐴 ) ) |
14 |
9 13
|
mpcom |
⊢ ( 𝑦 ∈ { 𝑧 ∈ ℝ ∣ - 𝑧 ∈ 𝐴 } → - 𝑦 ∈ 𝐴 ) |
15 |
|
breq1 |
⊢ ( 𝑏 = - 𝑦 → ( 𝑏 ≤ 𝑎 ↔ - 𝑦 ≤ 𝑎 ) ) |
16 |
15
|
rspcv |
⊢ ( - 𝑦 ∈ 𝐴 → ( ∀ 𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 → - 𝑦 ≤ 𝑎 ) ) |
17 |
14 16
|
syl |
⊢ ( 𝑦 ∈ { 𝑧 ∈ ℝ ∣ - 𝑧 ∈ 𝐴 } → ( ∀ 𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 → - 𝑦 ≤ 𝑎 ) ) |
18 |
17
|
adantl |
⊢ ( ( 𝑎 ∈ ℝ ∧ 𝑦 ∈ { 𝑧 ∈ ℝ ∣ - 𝑧 ∈ 𝐴 } ) → ( ∀ 𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 → - 𝑦 ≤ 𝑎 ) ) |
19 |
|
lenegcon1 |
⊢ ( ( 𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( - 𝑎 ≤ 𝑦 ↔ - 𝑦 ≤ 𝑎 ) ) |
20 |
9 19
|
sylan2 |
⊢ ( ( 𝑎 ∈ ℝ ∧ 𝑦 ∈ { 𝑧 ∈ ℝ ∣ - 𝑧 ∈ 𝐴 } ) → ( - 𝑎 ≤ 𝑦 ↔ - 𝑦 ≤ 𝑎 ) ) |
21 |
18 20
|
sylibrd |
⊢ ( ( 𝑎 ∈ ℝ ∧ 𝑦 ∈ { 𝑧 ∈ ℝ ∣ - 𝑧 ∈ 𝐴 } ) → ( ∀ 𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 → - 𝑎 ≤ 𝑦 ) ) |
22 |
21
|
ralrimdva |
⊢ ( 𝑎 ∈ ℝ → ( ∀ 𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 → ∀ 𝑦 ∈ { 𝑧 ∈ ℝ ∣ - 𝑧 ∈ 𝐴 } - 𝑎 ≤ 𝑦 ) ) |
23 |
|
breq1 |
⊢ ( 𝑥 = - 𝑎 → ( 𝑥 ≤ 𝑦 ↔ - 𝑎 ≤ 𝑦 ) ) |
24 |
23
|
ralbidv |
⊢ ( 𝑥 = - 𝑎 → ( ∀ 𝑦 ∈ { 𝑧 ∈ ℝ ∣ - 𝑧 ∈ 𝐴 } 𝑥 ≤ 𝑦 ↔ ∀ 𝑦 ∈ { 𝑧 ∈ ℝ ∣ - 𝑧 ∈ 𝐴 } - 𝑎 ≤ 𝑦 ) ) |
25 |
24
|
rspcev |
⊢ ( ( - 𝑎 ∈ ℝ ∧ ∀ 𝑦 ∈ { 𝑧 ∈ ℝ ∣ - 𝑧 ∈ 𝐴 } - 𝑎 ≤ 𝑦 ) → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ { 𝑧 ∈ ℝ ∣ - 𝑧 ∈ 𝐴 } 𝑥 ≤ 𝑦 ) |
26 |
8 22 25
|
syl6an |
⊢ ( 𝑎 ∈ ℝ → ( ∀ 𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ { 𝑧 ∈ ℝ ∣ - 𝑧 ∈ 𝐴 } 𝑥 ≤ 𝑦 ) ) |
27 |
26
|
rexlimiv |
⊢ ( ∃ 𝑎 ∈ ℝ ∀ 𝑏 ∈ 𝐴 𝑏 ≤ 𝑎 → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ { 𝑧 ∈ ℝ ∣ - 𝑧 ∈ 𝐴 } 𝑥 ≤ 𝑦 ) |
28 |
7 27
|
sylbir |
⊢ ( ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ { 𝑧 ∈ ℝ ∣ - 𝑧 ∈ 𝐴 } 𝑥 ≤ 𝑦 ) |