| Step |
Hyp |
Ref |
Expression |
| 1 |
|
wexp.1 |
|- T = { <. x , y >. | ( ( x e. ( A X. B ) /\ y e. ( A X. B ) ) /\ ( ( 1st ` x ) R ( 1st ` y ) \/ ( ( 1st ` x ) = ( 1st ` y ) /\ ( 2nd ` x ) S ( 2nd ` y ) ) ) ) } |
| 2 |
|
wefr |
|- ( R We A -> R Fr A ) |
| 3 |
|
wefr |
|- ( S We B -> S Fr B ) |
| 4 |
1
|
frxp |
|- ( ( R Fr A /\ S Fr B ) -> T Fr ( A X. B ) ) |
| 5 |
2 3 4
|
syl2an |
|- ( ( R We A /\ S We B ) -> T Fr ( A X. B ) ) |
| 6 |
|
weso |
|- ( R We A -> R Or A ) |
| 7 |
|
weso |
|- ( S We B -> S Or B ) |
| 8 |
1
|
soxp |
|- ( ( R Or A /\ S Or B ) -> T Or ( A X. B ) ) |
| 9 |
6 7 8
|
syl2an |
|- ( ( R We A /\ S We B ) -> T Or ( A X. B ) ) |
| 10 |
|
df-we |
|- ( T We ( A X. B ) <-> ( T Fr ( A X. B ) /\ T Or ( A X. B ) ) ) |
| 11 |
5 9 10
|
sylanbrc |
|- ( ( R We A /\ S We B ) -> T We ( A X. B ) ) |