Step |
Hyp |
Ref |
Expression |
1 |
|
2mpo0.o |
⊢ 𝑂 = ( 𝑥 ∈ 𝐴 , 𝑦 ∈ 𝐵 ↦ 𝐸 ) |
2 |
|
2mpo0.u |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 𝑂 𝑌 ) = ( 𝑠 ∈ 𝐶 , 𝑡 ∈ 𝐷 ↦ 𝐹 ) ) |
3 |
|
ianor |
⊢ ( ¬ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷 ) ) ↔ ( ¬ ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ) ∨ ¬ ( 𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷 ) ) ) |
4 |
1
|
mpondm0 |
⊢ ( ¬ ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 𝑂 𝑌 ) = ∅ ) |
5 |
4
|
oveqd |
⊢ ( ¬ ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑆 ( 𝑋 𝑂 𝑌 ) 𝑇 ) = ( 𝑆 ∅ 𝑇 ) ) |
6 |
|
0ov |
⊢ ( 𝑆 ∅ 𝑇 ) = ∅ |
7 |
5 6
|
eqtrdi |
⊢ ( ¬ ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑆 ( 𝑋 𝑂 𝑌 ) 𝑇 ) = ∅ ) |
8 |
|
notnotb |
⊢ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ) ↔ ¬ ¬ ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ) ) |
9 |
2
|
adantr |
⊢ ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷 ) ) → ( 𝑋 𝑂 𝑌 ) = ( 𝑠 ∈ 𝐶 , 𝑡 ∈ 𝐷 ↦ 𝐹 ) ) |
10 |
9
|
oveqd |
⊢ ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷 ) ) → ( 𝑆 ( 𝑋 𝑂 𝑌 ) 𝑇 ) = ( 𝑆 ( 𝑠 ∈ 𝐶 , 𝑡 ∈ 𝐷 ↦ 𝐹 ) 𝑇 ) ) |
11 |
|
eqid |
⊢ ( 𝑠 ∈ 𝐶 , 𝑡 ∈ 𝐷 ↦ 𝐹 ) = ( 𝑠 ∈ 𝐶 , 𝑡 ∈ 𝐷 ↦ 𝐹 ) |
12 |
11
|
mpondm0 |
⊢ ( ¬ ( 𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷 ) → ( 𝑆 ( 𝑠 ∈ 𝐶 , 𝑡 ∈ 𝐷 ↦ 𝐹 ) 𝑇 ) = ∅ ) |
13 |
12
|
adantl |
⊢ ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷 ) ) → ( 𝑆 ( 𝑠 ∈ 𝐶 , 𝑡 ∈ 𝐷 ↦ 𝐹 ) 𝑇 ) = ∅ ) |
14 |
10 13
|
eqtrd |
⊢ ( ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷 ) ) → ( 𝑆 ( 𝑋 𝑂 𝑌 ) 𝑇 ) = ∅ ) |
15 |
8 14
|
sylanbr |
⊢ ( ( ¬ ¬ ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ) ∧ ¬ ( 𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷 ) ) → ( 𝑆 ( 𝑋 𝑂 𝑌 ) 𝑇 ) = ∅ ) |
16 |
7 15
|
jaoi3 |
⊢ ( ( ¬ ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ) ∨ ¬ ( 𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷 ) ) → ( 𝑆 ( 𝑋 𝑂 𝑌 ) 𝑇 ) = ∅ ) |
17 |
3 16
|
sylbi |
⊢ ( ¬ ( ( 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑆 ∈ 𝐶 ∧ 𝑇 ∈ 𝐷 ) ) → ( 𝑆 ( 𝑋 𝑂 𝑌 ) 𝑇 ) = ∅ ) |