Description: Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 24-Jul-2021)
Ref | Expression | ||
---|---|---|---|
Assertion | eldisjs | |- ( R e. Disjs <-> ( ,~ `' R e. CnvRefRels /\ R e. Rels ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfdisjs | |- Disjs = { r e. Rels | ,~ `' r e. CnvRefRels } |
|
2 | cnveq | |- ( r = R -> `' r = `' R ) |
|
3 | 2 | cosseqd | |- ( r = R -> ,~ `' r = ,~ `' R ) |
4 | 3 | eleq1d | |- ( r = R -> ( ,~ `' r e. CnvRefRels <-> ,~ `' R e. CnvRefRels ) ) |
5 | 1 4 | rabeqel | |- ( R e. Disjs <-> ( ,~ `' R e. CnvRefRels /\ R e. Rels ) ) |