Step |
Hyp |
Ref |
Expression |
1 |
|
leweon.1 |
|- L = { <. x , y >. | ( ( x e. ( On X. On ) /\ y e. ( On X. On ) ) /\ ( ( 1st ` x ) e. ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) e. ( 2nd ` y ) ) ) ) } |
2 |
|
epweon |
|- _E We On |
3 |
|
fvex |
|- ( 1st ` y ) e. _V |
4 |
3
|
epeli |
|- ( ( 1st ` x ) _E ( 1st ` y ) <-> ( 1st ` x ) e. ( 1st ` y ) ) |
5 |
|
fvex |
|- ( 2nd ` y ) e. _V |
6 |
5
|
epeli |
|- ( ( 2nd ` x ) _E ( 2nd ` y ) <-> ( 2nd ` x ) e. ( 2nd ` y ) ) |
7 |
6
|
anbi2i |
|- ( ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) _E ( 2nd ` y ) ) <-> ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) e. ( 2nd ` y ) ) ) |
8 |
4 7
|
orbi12i |
|- ( ( ( 1st ` x ) _E ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) _E ( 2nd ` y ) ) ) <-> ( ( 1st ` x ) e. ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) e. ( 2nd ` y ) ) ) ) |
9 |
8
|
anbi2i |
|- ( ( ( x e. ( On X. On ) /\ y e. ( On X. On ) ) /\ ( ( 1st ` x ) _E ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) _E ( 2nd ` y ) ) ) ) <-> ( ( x e. ( On X. On ) /\ y e. ( On X. On ) ) /\ ( ( 1st ` x ) e. ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) e. ( 2nd ` y ) ) ) ) ) |
10 |
9
|
opabbii |
|- { <. x , y >. | ( ( x e. ( On X. On ) /\ y e. ( On X. On ) ) /\ ( ( 1st ` x ) _E ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) _E ( 2nd ` y ) ) ) ) } = { <. x , y >. | ( ( x e. ( On X. On ) /\ y e. ( On X. On ) ) /\ ( ( 1st ` x ) e. ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) e. ( 2nd ` y ) ) ) ) } |
11 |
1 10
|
eqtr4i |
|- L = { <. x , y >. | ( ( x e. ( On X. On ) /\ y e. ( On X. On ) ) /\ ( ( 1st ` x ) _E ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) _E ( 2nd ` y ) ) ) ) } |
12 |
11
|
wexp |
|- ( ( _E We On /\ _E We On ) -> L We ( On X. On ) ) |
13 |
2 2 12
|
mp2an |
|- L We ( On X. On ) |