Step |
Hyp |
Ref |
Expression |
1 |
|
sscon |
⊢ ( 𝐴 ⊆ 𝐵 → ( 𝐶 ∖ 𝐵 ) ⊆ ( 𝐶 ∖ 𝐴 ) ) |
2 |
|
sscon |
⊢ ( ( 𝐶 ∖ 𝐵 ) ⊆ ( 𝐶 ∖ 𝐴 ) → ( 𝐶 ∖ ( 𝐶 ∖ 𝐴 ) ) ⊆ ( 𝐶 ∖ ( 𝐶 ∖ 𝐵 ) ) ) |
3 |
|
dfss4 |
⊢ ( 𝐴 ⊆ 𝐶 ↔ ( 𝐶 ∖ ( 𝐶 ∖ 𝐴 ) ) = 𝐴 ) |
4 |
3
|
biimpi |
⊢ ( 𝐴 ⊆ 𝐶 → ( 𝐶 ∖ ( 𝐶 ∖ 𝐴 ) ) = 𝐴 ) |
5 |
4
|
adantr |
⊢ ( ( 𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶 ) → ( 𝐶 ∖ ( 𝐶 ∖ 𝐴 ) ) = 𝐴 ) |
6 |
|
dfss4 |
⊢ ( 𝐵 ⊆ 𝐶 ↔ ( 𝐶 ∖ ( 𝐶 ∖ 𝐵 ) ) = 𝐵 ) |
7 |
6
|
biimpi |
⊢ ( 𝐵 ⊆ 𝐶 → ( 𝐶 ∖ ( 𝐶 ∖ 𝐵 ) ) = 𝐵 ) |
8 |
7
|
adantl |
⊢ ( ( 𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶 ) → ( 𝐶 ∖ ( 𝐶 ∖ 𝐵 ) ) = 𝐵 ) |
9 |
5 8
|
sseq12d |
⊢ ( ( 𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶 ) → ( ( 𝐶 ∖ ( 𝐶 ∖ 𝐴 ) ) ⊆ ( 𝐶 ∖ ( 𝐶 ∖ 𝐵 ) ) ↔ 𝐴 ⊆ 𝐵 ) ) |
10 |
2 9
|
syl5ib |
⊢ ( ( 𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶 ) → ( ( 𝐶 ∖ 𝐵 ) ⊆ ( 𝐶 ∖ 𝐴 ) → 𝐴 ⊆ 𝐵 ) ) |
11 |
1 10
|
impbid2 |
⊢ ( ( 𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐶 ) → ( 𝐴 ⊆ 𝐵 ↔ ( 𝐶 ∖ 𝐵 ) ⊆ ( 𝐶 ∖ 𝐴 ) ) ) |