Step |
Hyp |
Ref |
Expression |
1 |
|
elin |
⊢ ( 𝐵 ∈ ( [ 𝐴 ] ≀ 𝑅 ∩ [ 𝐶 ] ≀ 𝑅 ) ↔ ( 𝐵 ∈ [ 𝐴 ] ≀ 𝑅 ∧ 𝐵 ∈ [ 𝐶 ] ≀ 𝑅 ) ) |
2 |
|
relcoss |
⊢ Rel ≀ 𝑅 |
3 |
|
relelec |
⊢ ( Rel ≀ 𝑅 → ( 𝐵 ∈ [ 𝐴 ] ≀ 𝑅 ↔ 𝐴 ≀ 𝑅 𝐵 ) ) |
4 |
2 3
|
ax-mp |
⊢ ( 𝐵 ∈ [ 𝐴 ] ≀ 𝑅 ↔ 𝐴 ≀ 𝑅 𝐵 ) |
5 |
|
relelec |
⊢ ( Rel ≀ 𝑅 → ( 𝐵 ∈ [ 𝐶 ] ≀ 𝑅 ↔ 𝐶 ≀ 𝑅 𝐵 ) ) |
6 |
2 5
|
ax-mp |
⊢ ( 𝐵 ∈ [ 𝐶 ] ≀ 𝑅 ↔ 𝐶 ≀ 𝑅 𝐵 ) |
7 |
4 6
|
anbi12i |
⊢ ( ( 𝐵 ∈ [ 𝐴 ] ≀ 𝑅 ∧ 𝐵 ∈ [ 𝐶 ] ≀ 𝑅 ) ↔ ( 𝐴 ≀ 𝑅 𝐵 ∧ 𝐶 ≀ 𝑅 𝐵 ) ) |
8 |
1 7
|
bitri |
⊢ ( 𝐵 ∈ ( [ 𝐴 ] ≀ 𝑅 ∩ [ 𝐶 ] ≀ 𝑅 ) ↔ ( 𝐴 ≀ 𝑅 𝐵 ∧ 𝐶 ≀ 𝑅 𝐵 ) ) |
9 |
|
brcosscnvcoss |
⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( 𝐵 ≀ 𝑅 𝐶 ↔ 𝐶 ≀ 𝑅 𝐵 ) ) |
10 |
9
|
anbi2d |
⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( ( 𝐴 ≀ 𝑅 𝐵 ∧ 𝐵 ≀ 𝑅 𝐶 ) ↔ ( 𝐴 ≀ 𝑅 𝐵 ∧ 𝐶 ≀ 𝑅 𝐵 ) ) ) |
11 |
8 10
|
bitr4id |
⊢ ( ( 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑊 ) → ( 𝐵 ∈ ( [ 𝐴 ] ≀ 𝑅 ∩ [ 𝐶 ] ≀ 𝑅 ) ↔ ( 𝐴 ≀ 𝑅 𝐵 ∧ 𝐵 ≀ 𝑅 𝐶 ) ) ) |