Step |
Hyp |
Ref |
Expression |
1 |
|
disjexc.1 |
⊢ ( 𝑥 = 𝑦 → 𝐴 = 𝐵 ) |
2 |
1
|
imim2i |
⊢ ( ( ∃ 𝑧 ( 𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) → 𝑥 = 𝑦 ) → ( ∃ 𝑧 ( 𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) → 𝐴 = 𝐵 ) ) |
3 |
|
orcom |
⊢ ( ( 𝐴 = 𝐵 ∨ ¬ ∃ 𝑧 ( 𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ) ↔ ( ¬ ∃ 𝑧 ( 𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ∨ 𝐴 = 𝐵 ) ) |
4 |
|
df-in |
⊢ ( 𝐴 ∩ 𝐵 ) = { 𝑧 ∣ ( 𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) } |
5 |
4
|
neeq1i |
⊢ ( ( 𝐴 ∩ 𝐵 ) ≠ ∅ ↔ { 𝑧 ∣ ( 𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) } ≠ ∅ ) |
6 |
|
abn0 |
⊢ ( { 𝑧 ∣ ( 𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) } ≠ ∅ ↔ ∃ 𝑧 ( 𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ) |
7 |
5 6
|
bitr2i |
⊢ ( ∃ 𝑧 ( 𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ↔ ( 𝐴 ∩ 𝐵 ) ≠ ∅ ) |
8 |
7
|
necon2bbii |
⊢ ( ( 𝐴 ∩ 𝐵 ) = ∅ ↔ ¬ ∃ 𝑧 ( 𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ) |
9 |
8
|
orbi2i |
⊢ ( ( 𝐴 = 𝐵 ∨ ( 𝐴 ∩ 𝐵 ) = ∅ ) ↔ ( 𝐴 = 𝐵 ∨ ¬ ∃ 𝑧 ( 𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ) ) |
10 |
|
imor |
⊢ ( ( ∃ 𝑧 ( 𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) → 𝐴 = 𝐵 ) ↔ ( ¬ ∃ 𝑧 ( 𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) ∨ 𝐴 = 𝐵 ) ) |
11 |
3 9 10
|
3bitr4i |
⊢ ( ( 𝐴 = 𝐵 ∨ ( 𝐴 ∩ 𝐵 ) = ∅ ) ↔ ( ∃ 𝑧 ( 𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) → 𝐴 = 𝐵 ) ) |
12 |
2 11
|
sylibr |
⊢ ( ( ∃ 𝑧 ( 𝑧 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵 ) → 𝑥 = 𝑦 ) → ( 𝐴 = 𝐵 ∨ ( 𝐴 ∩ 𝐵 ) = ∅ ) ) |