Step |
Hyp |
Ref |
Expression |
1 |
|
lerel |
⊢ Rel ≤ |
2 |
|
ltrelxr |
⊢ < ⊆ ( ℝ* × ℝ* ) |
3 |
|
idssxp |
⊢ ( I ↾ ℝ* ) ⊆ ( ℝ* × ℝ* ) |
4 |
2 3
|
unssi |
⊢ ( < ∪ ( I ↾ ℝ* ) ) ⊆ ( ℝ* × ℝ* ) |
5 |
|
relxp |
⊢ Rel ( ℝ* × ℝ* ) |
6 |
|
relss |
⊢ ( ( < ∪ ( I ↾ ℝ* ) ) ⊆ ( ℝ* × ℝ* ) → ( Rel ( ℝ* × ℝ* ) → Rel ( < ∪ ( I ↾ ℝ* ) ) ) ) |
7 |
4 5 6
|
mp2 |
⊢ Rel ( < ∪ ( I ↾ ℝ* ) ) |
8 |
|
lerelxr |
⊢ ≤ ⊆ ( ℝ* × ℝ* ) |
9 |
8
|
brel |
⊢ ( 𝑥 ≤ 𝑦 → ( 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ) |
10 |
4
|
brel |
⊢ ( 𝑥 ( < ∪ ( I ↾ ℝ* ) ) 𝑦 → ( 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ) |
11 |
|
xrleloe |
⊢ ( ( 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ( 𝑥 ≤ 𝑦 ↔ ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ) ) ) |
12 |
|
resieq |
⊢ ( ( 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ( 𝑥 ( I ↾ ℝ* ) 𝑦 ↔ 𝑥 = 𝑦 ) ) |
13 |
12
|
orbi2d |
⊢ ( ( 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ( ( 𝑥 < 𝑦 ∨ 𝑥 ( I ↾ ℝ* ) 𝑦 ) ↔ ( 𝑥 < 𝑦 ∨ 𝑥 = 𝑦 ) ) ) |
14 |
11 13
|
bitr4d |
⊢ ( ( 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ( 𝑥 ≤ 𝑦 ↔ ( 𝑥 < 𝑦 ∨ 𝑥 ( I ↾ ℝ* ) 𝑦 ) ) ) |
15 |
|
brun |
⊢ ( 𝑥 ( < ∪ ( I ↾ ℝ* ) ) 𝑦 ↔ ( 𝑥 < 𝑦 ∨ 𝑥 ( I ↾ ℝ* ) 𝑦 ) ) |
16 |
14 15
|
bitr4di |
⊢ ( ( 𝑥 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ( 𝑥 ≤ 𝑦 ↔ 𝑥 ( < ∪ ( I ↾ ℝ* ) ) 𝑦 ) ) |
17 |
9 10 16
|
pm5.21nii |
⊢ ( 𝑥 ≤ 𝑦 ↔ 𝑥 ( < ∪ ( I ↾ ℝ* ) ) 𝑦 ) |
18 |
1 7 17
|
eqbrriv |
⊢ ≤ = ( < ∪ ( I ↾ ℝ* ) ) |