Step |
Hyp |
Ref |
Expression |
1 |
|
difss |
⊢ ( ( 𝐶 × 𝐷 ) ∖ ( 𝐴 × 𝐵 ) ) ⊆ ( 𝐶 × 𝐷 ) |
2 |
|
relxp |
⊢ Rel ( 𝐶 × 𝐷 ) |
3 |
|
relss |
⊢ ( ( ( 𝐶 × 𝐷 ) ∖ ( 𝐴 × 𝐵 ) ) ⊆ ( 𝐶 × 𝐷 ) → ( Rel ( 𝐶 × 𝐷 ) → Rel ( ( 𝐶 × 𝐷 ) ∖ ( 𝐴 × 𝐵 ) ) ) ) |
4 |
1 2 3
|
mp2 |
⊢ Rel ( ( 𝐶 × 𝐷 ) ∖ ( 𝐴 × 𝐵 ) ) |
5 |
|
relxp |
⊢ Rel ( ( 𝐶 ∖ 𝐴 ) × 𝐷 ) |
6 |
|
relxp |
⊢ Rel ( 𝐶 × ( 𝐷 ∖ 𝐵 ) ) |
7 |
|
relun |
⊢ ( Rel ( ( ( 𝐶 ∖ 𝐴 ) × 𝐷 ) ∪ ( 𝐶 × ( 𝐷 ∖ 𝐵 ) ) ) ↔ ( Rel ( ( 𝐶 ∖ 𝐴 ) × 𝐷 ) ∧ Rel ( 𝐶 × ( 𝐷 ∖ 𝐵 ) ) ) ) |
8 |
5 6 7
|
mpbir2an |
⊢ Rel ( ( ( 𝐶 ∖ 𝐴 ) × 𝐷 ) ∪ ( 𝐶 × ( 𝐷 ∖ 𝐵 ) ) ) |
9 |
|
ianor |
⊢ ( ¬ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ↔ ( ¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 ∈ 𝐵 ) ) |
10 |
9
|
anbi2i |
⊢ ( ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ ¬ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) ↔ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ ( ¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 ∈ 𝐵 ) ) ) |
11 |
|
andi |
⊢ ( ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ ( ¬ 𝑥 ∈ 𝐴 ∨ ¬ 𝑦 ∈ 𝐵 ) ) ↔ ( ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ ¬ 𝑥 ∈ 𝐴 ) ∨ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ ¬ 𝑦 ∈ 𝐵 ) ) ) |
12 |
10 11
|
bitri |
⊢ ( ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ ¬ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) ↔ ( ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ ¬ 𝑥 ∈ 𝐴 ) ∨ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ ¬ 𝑦 ∈ 𝐵 ) ) ) |
13 |
|
opelxp |
⊢ ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐶 × 𝐷 ) ↔ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ) |
14 |
|
opelxp |
⊢ ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐴 × 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) |
15 |
14
|
notbii |
⊢ ( ¬ 〈 𝑥 , 𝑦 〉 ∈ ( 𝐴 × 𝐵 ) ↔ ¬ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) |
16 |
13 15
|
anbi12i |
⊢ ( ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐶 × 𝐷 ) ∧ ¬ 〈 𝑥 , 𝑦 〉 ∈ ( 𝐴 × 𝐵 ) ) ↔ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ ¬ ( 𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵 ) ) ) |
17 |
|
opelxp |
⊢ ( 〈 𝑥 , 𝑦 〉 ∈ ( ( 𝐶 ∖ 𝐴 ) × 𝐷 ) ↔ ( 𝑥 ∈ ( 𝐶 ∖ 𝐴 ) ∧ 𝑦 ∈ 𝐷 ) ) |
18 |
|
eldif |
⊢ ( 𝑥 ∈ ( 𝐶 ∖ 𝐴 ) ↔ ( 𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴 ) ) |
19 |
18
|
anbi1i |
⊢ ( ( 𝑥 ∈ ( 𝐶 ∖ 𝐴 ) ∧ 𝑦 ∈ 𝐷 ) ↔ ( ( 𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐷 ) ) |
20 |
|
an32 |
⊢ ( ( ( 𝑥 ∈ 𝐶 ∧ ¬ 𝑥 ∈ 𝐴 ) ∧ 𝑦 ∈ 𝐷 ) ↔ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ ¬ 𝑥 ∈ 𝐴 ) ) |
21 |
19 20
|
bitri |
⊢ ( ( 𝑥 ∈ ( 𝐶 ∖ 𝐴 ) ∧ 𝑦 ∈ 𝐷 ) ↔ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ ¬ 𝑥 ∈ 𝐴 ) ) |
22 |
17 21
|
bitri |
⊢ ( 〈 𝑥 , 𝑦 〉 ∈ ( ( 𝐶 ∖ 𝐴 ) × 𝐷 ) ↔ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ ¬ 𝑥 ∈ 𝐴 ) ) |
23 |
|
eldif |
⊢ ( 𝑦 ∈ ( 𝐷 ∖ 𝐵 ) ↔ ( 𝑦 ∈ 𝐷 ∧ ¬ 𝑦 ∈ 𝐵 ) ) |
24 |
23
|
anbi2i |
⊢ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ ( 𝐷 ∖ 𝐵 ) ) ↔ ( 𝑥 ∈ 𝐶 ∧ ( 𝑦 ∈ 𝐷 ∧ ¬ 𝑦 ∈ 𝐵 ) ) ) |
25 |
|
opelxp |
⊢ ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐶 × ( 𝐷 ∖ 𝐵 ) ) ↔ ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ ( 𝐷 ∖ 𝐵 ) ) ) |
26 |
|
anass |
⊢ ( ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ ¬ 𝑦 ∈ 𝐵 ) ↔ ( 𝑥 ∈ 𝐶 ∧ ( 𝑦 ∈ 𝐷 ∧ ¬ 𝑦 ∈ 𝐵 ) ) ) |
27 |
24 25 26
|
3bitr4i |
⊢ ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐶 × ( 𝐷 ∖ 𝐵 ) ) ↔ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ ¬ 𝑦 ∈ 𝐵 ) ) |
28 |
22 27
|
orbi12i |
⊢ ( ( 〈 𝑥 , 𝑦 〉 ∈ ( ( 𝐶 ∖ 𝐴 ) × 𝐷 ) ∨ 〈 𝑥 , 𝑦 〉 ∈ ( 𝐶 × ( 𝐷 ∖ 𝐵 ) ) ) ↔ ( ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ ¬ 𝑥 ∈ 𝐴 ) ∨ ( ( 𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐷 ) ∧ ¬ 𝑦 ∈ 𝐵 ) ) ) |
29 |
12 16 28
|
3bitr4i |
⊢ ( ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐶 × 𝐷 ) ∧ ¬ 〈 𝑥 , 𝑦 〉 ∈ ( 𝐴 × 𝐵 ) ) ↔ ( 〈 𝑥 , 𝑦 〉 ∈ ( ( 𝐶 ∖ 𝐴 ) × 𝐷 ) ∨ 〈 𝑥 , 𝑦 〉 ∈ ( 𝐶 × ( 𝐷 ∖ 𝐵 ) ) ) ) |
30 |
|
eldif |
⊢ ( 〈 𝑥 , 𝑦 〉 ∈ ( ( 𝐶 × 𝐷 ) ∖ ( 𝐴 × 𝐵 ) ) ↔ ( 〈 𝑥 , 𝑦 〉 ∈ ( 𝐶 × 𝐷 ) ∧ ¬ 〈 𝑥 , 𝑦 〉 ∈ ( 𝐴 × 𝐵 ) ) ) |
31 |
|
elun |
⊢ ( 〈 𝑥 , 𝑦 〉 ∈ ( ( ( 𝐶 ∖ 𝐴 ) × 𝐷 ) ∪ ( 𝐶 × ( 𝐷 ∖ 𝐵 ) ) ) ↔ ( 〈 𝑥 , 𝑦 〉 ∈ ( ( 𝐶 ∖ 𝐴 ) × 𝐷 ) ∨ 〈 𝑥 , 𝑦 〉 ∈ ( 𝐶 × ( 𝐷 ∖ 𝐵 ) ) ) ) |
32 |
29 30 31
|
3bitr4i |
⊢ ( 〈 𝑥 , 𝑦 〉 ∈ ( ( 𝐶 × 𝐷 ) ∖ ( 𝐴 × 𝐵 ) ) ↔ 〈 𝑥 , 𝑦 〉 ∈ ( ( ( 𝐶 ∖ 𝐴 ) × 𝐷 ) ∪ ( 𝐶 × ( 𝐷 ∖ 𝐵 ) ) ) ) |
33 |
4 8 32
|
eqrelriiv |
⊢ ( ( 𝐶 × 𝐷 ) ∖ ( 𝐴 × 𝐵 ) ) = ( ( ( 𝐶 ∖ 𝐴 ) × 𝐷 ) ∪ ( 𝐶 × ( 𝐷 ∖ 𝐵 ) ) ) |