Step |
Hyp |
Ref |
Expression |
1 |
|
disjiun2.1 |
⊢ ( 𝜑 → Disj 𝑥 ∈ 𝐴 𝐵 ) |
2 |
|
disjiun2.2 |
⊢ ( 𝜑 → 𝐶 ⊆ 𝐴 ) |
3 |
|
disjiun2.3 |
⊢ ( 𝜑 → 𝐷 ∈ ( 𝐴 ∖ 𝐶 ) ) |
4 |
|
disjiun2.4 |
⊢ ( 𝑥 = 𝐷 → 𝐵 = 𝐸 ) |
5 |
4
|
iunxsng |
⊢ ( 𝐷 ∈ ( 𝐴 ∖ 𝐶 ) → ∪ 𝑥 ∈ { 𝐷 } 𝐵 = 𝐸 ) |
6 |
3 5
|
syl |
⊢ ( 𝜑 → ∪ 𝑥 ∈ { 𝐷 } 𝐵 = 𝐸 ) |
7 |
6
|
ineq2d |
⊢ ( 𝜑 → ( ∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ { 𝐷 } 𝐵 ) = ( ∪ 𝑥 ∈ 𝐶 𝐵 ∩ 𝐸 ) ) |
8 |
|
eldifi |
⊢ ( 𝐷 ∈ ( 𝐴 ∖ 𝐶 ) → 𝐷 ∈ 𝐴 ) |
9 |
|
snssi |
⊢ ( 𝐷 ∈ 𝐴 → { 𝐷 } ⊆ 𝐴 ) |
10 |
3 8 9
|
3syl |
⊢ ( 𝜑 → { 𝐷 } ⊆ 𝐴 ) |
11 |
3
|
eldifbd |
⊢ ( 𝜑 → ¬ 𝐷 ∈ 𝐶 ) |
12 |
|
disjsn |
⊢ ( ( 𝐶 ∩ { 𝐷 } ) = ∅ ↔ ¬ 𝐷 ∈ 𝐶 ) |
13 |
11 12
|
sylibr |
⊢ ( 𝜑 → ( 𝐶 ∩ { 𝐷 } ) = ∅ ) |
14 |
|
disjiun |
⊢ ( ( Disj 𝑥 ∈ 𝐴 𝐵 ∧ ( 𝐶 ⊆ 𝐴 ∧ { 𝐷 } ⊆ 𝐴 ∧ ( 𝐶 ∩ { 𝐷 } ) = ∅ ) ) → ( ∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ { 𝐷 } 𝐵 ) = ∅ ) |
15 |
1 2 10 13 14
|
syl13anc |
⊢ ( 𝜑 → ( ∪ 𝑥 ∈ 𝐶 𝐵 ∩ ∪ 𝑥 ∈ { 𝐷 } 𝐵 ) = ∅ ) |
16 |
7 15
|
eqtr3d |
⊢ ( 𝜑 → ( ∪ 𝑥 ∈ 𝐶 𝐵 ∩ 𝐸 ) = ∅ ) |