Step |
Hyp |
Ref |
Expression |
1 |
|
cosscnvex |
|- ( R e. V -> ,~ `' R e. _V ) |
2 |
|
elcnvrefrelsrel |
|- ( ,~ `' R e. _V -> ( ,~ `' R e. CnvRefRels <-> CnvRefRel ,~ `' R ) ) |
3 |
1 2
|
syl |
|- ( R e. V -> ( ,~ `' R e. CnvRefRels <-> CnvRefRel ,~ `' R ) ) |
4 |
|
elrelsrel |
|- ( R e. V -> ( R e. Rels <-> Rel R ) ) |
5 |
3 4
|
anbi12d |
|- ( R e. V -> ( ( ,~ `' R e. CnvRefRels /\ R e. Rels ) <-> ( CnvRefRel ,~ `' R /\ Rel R ) ) ) |
6 |
|
eldisjs |
|- ( R e. Disjs <-> ( ,~ `' R e. CnvRefRels /\ R e. Rels ) ) |
7 |
|
df-disjALTV |
|- ( Disj R <-> ( CnvRefRel ,~ `' R /\ Rel R ) ) |
8 |
5 6 7
|
3bitr4g |
|- ( R e. V -> ( R e. Disjs <-> Disj R ) ) |