Step |
Hyp |
Ref |
Expression |
1 |
|
simprl |
⊢ ( ( [⊊] Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → 𝐵 ∈ 𝐴 ) |
2 |
|
df-ss |
⊢ ( 𝐵 ⊆ 𝐶 ↔ ( 𝐵 ∩ 𝐶 ) = 𝐵 ) |
3 |
|
eleq1 |
⊢ ( ( 𝐵 ∩ 𝐶 ) = 𝐵 → ( ( 𝐵 ∩ 𝐶 ) ∈ 𝐴 ↔ 𝐵 ∈ 𝐴 ) ) |
4 |
2 3
|
sylbi |
⊢ ( 𝐵 ⊆ 𝐶 → ( ( 𝐵 ∩ 𝐶 ) ∈ 𝐴 ↔ 𝐵 ∈ 𝐴 ) ) |
5 |
1 4
|
syl5ibrcom |
⊢ ( ( [⊊] Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ( 𝐵 ⊆ 𝐶 → ( 𝐵 ∩ 𝐶 ) ∈ 𝐴 ) ) |
6 |
|
simprr |
⊢ ( ( [⊊] Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → 𝐶 ∈ 𝐴 ) |
7 |
|
sseqin2 |
⊢ ( 𝐶 ⊆ 𝐵 ↔ ( 𝐵 ∩ 𝐶 ) = 𝐶 ) |
8 |
|
eleq1 |
⊢ ( ( 𝐵 ∩ 𝐶 ) = 𝐶 → ( ( 𝐵 ∩ 𝐶 ) ∈ 𝐴 ↔ 𝐶 ∈ 𝐴 ) ) |
9 |
7 8
|
sylbi |
⊢ ( 𝐶 ⊆ 𝐵 → ( ( 𝐵 ∩ 𝐶 ) ∈ 𝐴 ↔ 𝐶 ∈ 𝐴 ) ) |
10 |
6 9
|
syl5ibrcom |
⊢ ( ( [⊊] Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ( 𝐶 ⊆ 𝐵 → ( 𝐵 ∩ 𝐶 ) ∈ 𝐴 ) ) |
11 |
|
sorpssi |
⊢ ( ( [⊊] Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ( 𝐵 ⊆ 𝐶 ∨ 𝐶 ⊆ 𝐵 ) ) |
12 |
5 10 11
|
mpjaod |
⊢ ( ( [⊊] Or 𝐴 ∧ ( 𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ) ) → ( 𝐵 ∩ 𝐶 ) ∈ 𝐴 ) |