Step |
Hyp |
Ref |
Expression |
1 |
|
reli |
|- Rel _I |
2 |
|
relsset |
|- Rel SSet |
3 |
|
relin1 |
|- ( Rel SSet -> Rel ( SSet i^i `' SSet ) ) |
4 |
2 3
|
ax-mp |
|- Rel ( SSet i^i `' SSet ) |
5 |
|
eqss |
|- ( y = z <-> ( y C_ z /\ z C_ y ) ) |
6 |
|
vex |
|- z e. _V |
7 |
6
|
ideq |
|- ( y _I z <-> y = z ) |
8 |
|
brin |
|- ( y ( SSet i^i `' SSet ) z <-> ( y SSet z /\ y `' SSet z ) ) |
9 |
6
|
brsset |
|- ( y SSet z <-> y C_ z ) |
10 |
|
vex |
|- y e. _V |
11 |
10 6
|
brcnv |
|- ( y `' SSet z <-> z SSet y ) |
12 |
10
|
brsset |
|- ( z SSet y <-> z C_ y ) |
13 |
11 12
|
bitri |
|- ( y `' SSet z <-> z C_ y ) |
14 |
9 13
|
anbi12i |
|- ( ( y SSet z /\ y `' SSet z ) <-> ( y C_ z /\ z C_ y ) ) |
15 |
8 14
|
bitri |
|- ( y ( SSet i^i `' SSet ) z <-> ( y C_ z /\ z C_ y ) ) |
16 |
5 7 15
|
3bitr4i |
|- ( y _I z <-> y ( SSet i^i `' SSet ) z ) |
17 |
1 4 16
|
eqbrriv |
|- _I = ( SSet i^i `' SSet ) |