Metamath Proof Explorer

Theorem eldisjs2

Description: Elementhood in the class of disjoints. (Contributed by Peter Mazsa, 5-Sep-2021)

Ref Expression
Assertion eldisjs2 
|- ( R e. Disjs <-> ( ,~ `' R C_ _I /\ R e. Rels ) )

Proof

Step Hyp Ref Expression
1 eldisjs 
 |-  ( R e. Disjs <-> ( ,~ `' R e. CnvRefRels /\ R e. Rels ) )
2 cosselcnvrefrels2 
 |-  ( ,~ `' R e. CnvRefRels <-> ( ,~ `' R C_ _I /\ ,~ `' R e. Rels ) )
3 cosscnvelrels 
 |-  ( R e. Rels -> ,~ `' R e. Rels )
4 3 biantrud 
 |-  ( R e. Rels -> ( ,~ `' R C_ _I <-> ( ,~ `' R C_ _I /\ ,~ `' R e. Rels ) ) )
5 2 4 bitr4id 
 |-  ( R e. Rels -> ( ,~ `' R e. CnvRefRels <-> ,~ `' R C_ _I ) )
6 5 pm5.32ri 
 |-  ( ( ,~ `' R e. CnvRefRels /\ R e. Rels ) <-> ( ,~ `' R C_ _I /\ R e. Rels ) )
7 1 6 bitri 
 |-  ( R e. Disjs <-> ( ,~ `' R C_ _I /\ R e. Rels ) )