Step |
Hyp |
Ref |
Expression |
1 |
|
bj-inexeqex |
⊢ ( ( ( 𝐴 ∩ 𝐵 ) ∈ 𝑉 ∧ 𝐴 = 𝐵 ) → ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ) |
2 |
1
|
ex |
⊢ ( ( 𝐴 ∩ 𝐵 ) ∈ 𝑉 → ( 𝐴 = 𝐵 → ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ) ) |
3 |
|
bj-opelidb |
⊢ ( 〈 𝐴 , 𝐵 〉 ∈ I ↔ ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ∧ 𝐴 = 𝐵 ) ) |
4 |
|
simpr |
⊢ ( ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ∧ 𝐴 = 𝐵 ) → 𝐴 = 𝐵 ) |
5 |
|
ancr |
⊢ ( ( 𝐴 = 𝐵 → ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ) → ( 𝐴 = 𝐵 → ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ∧ 𝐴 = 𝐵 ) ) ) |
6 |
4 5
|
impbid2 |
⊢ ( ( 𝐴 = 𝐵 → ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ) → ( ( ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ∧ 𝐴 = 𝐵 ) ↔ 𝐴 = 𝐵 ) ) |
7 |
3 6
|
syl5bb |
⊢ ( ( 𝐴 = 𝐵 → ( 𝐴 ∈ V ∧ 𝐵 ∈ V ) ) → ( 〈 𝐴 , 𝐵 〉 ∈ I ↔ 𝐴 = 𝐵 ) ) |
8 |
2 7
|
syl |
⊢ ( ( 𝐴 ∩ 𝐵 ) ∈ 𝑉 → ( 〈 𝐴 , 𝐵 〉 ∈ I ↔ 𝐴 = 𝐵 ) ) |