Metamath Proof Explorer

Theorem eldisjs4

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

Ref Expression
Assertion eldisjs4 
|- ( R e. Disjs <-> ( A. x E* u u R x /\ R e. Rels ) )

Proof

Step Hyp Ref Expression
1 eldisjs2 
 |-  ( R e. Disjs <-> ( ,~ `' R C_ _I /\ R e. Rels ) )
2 cosscnvssid4 
 |-  ( ,~ `' R C_ _I <-> A. x E* u u R x )
3 2 anbi1i 
 |-  ( ( ,~ `' R C_ _I /\ R e. Rels ) <-> ( A. x E* u u R x /\ R e. Rels ) )
4 1 3 bitri 
 |-  ( R e. Disjs <-> ( A. x E* u u R x /\ R e. Rels ) )