Step |
Hyp |
Ref |
Expression |
1 |
|
reli |
⊢ Rel I |
2 |
|
relsset |
⊢ Rel SSet |
3 |
|
relin1 |
⊢ ( Rel SSet → Rel ( SSet ∩ ◡ SSet ) ) |
4 |
2 3
|
ax-mp |
⊢ Rel ( SSet ∩ ◡ SSet ) |
5 |
|
eqss |
⊢ ( 𝑦 = 𝑧 ↔ ( 𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ) |
6 |
|
vex |
⊢ 𝑧 ∈ V |
7 |
6
|
ideq |
⊢ ( 𝑦 I 𝑧 ↔ 𝑦 = 𝑧 ) |
8 |
|
brin |
⊢ ( 𝑦 ( SSet ∩ ◡ SSet ) 𝑧 ↔ ( 𝑦 SSet 𝑧 ∧ 𝑦 ◡ SSet 𝑧 ) ) |
9 |
6
|
brsset |
⊢ ( 𝑦 SSet 𝑧 ↔ 𝑦 ⊆ 𝑧 ) |
10 |
|
vex |
⊢ 𝑦 ∈ V |
11 |
10 6
|
brcnv |
⊢ ( 𝑦 ◡ SSet 𝑧 ↔ 𝑧 SSet 𝑦 ) |
12 |
10
|
brsset |
⊢ ( 𝑧 SSet 𝑦 ↔ 𝑧 ⊆ 𝑦 ) |
13 |
11 12
|
bitri |
⊢ ( 𝑦 ◡ SSet 𝑧 ↔ 𝑧 ⊆ 𝑦 ) |
14 |
9 13
|
anbi12i |
⊢ ( ( 𝑦 SSet 𝑧 ∧ 𝑦 ◡ SSet 𝑧 ) ↔ ( 𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ) |
15 |
8 14
|
bitri |
⊢ ( 𝑦 ( SSet ∩ ◡ SSet ) 𝑧 ↔ ( 𝑦 ⊆ 𝑧 ∧ 𝑧 ⊆ 𝑦 ) ) |
16 |
5 7 15
|
3bitr4i |
⊢ ( 𝑦 I 𝑧 ↔ 𝑦 ( SSet ∩ ◡ SSet ) 𝑧 ) |
17 |
1 4 16
|
eqbrriv |
⊢ I = ( SSet ∩ ◡ SSet ) |