Theorem eldisjs3

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

	Ref	Expression
Assertion	eldisjs3	|- ( R e. Disjs <-> ( A. u A. v A. x ( ( u R x /\ v R x ) -> u = v ) /\ R e. Rels ) )

Proof

Step	Hyp	Ref	Expression
1		eldisjs2	|- ( R e. Disjs <-> ( ,~ `' R C_ _I /\ R e. Rels ) )
2		cosscnvssid3	|- ( ,~ `' R C_ _I <-> A. u A. v A. x ( ( u R x /\ v R x ) -> u = v ) )
3	2	anbi1i	|- ( ( ,~ `' R C_ _I /\ R e. Rels ) <-> ( A. u A. v A. x ( ( u R x /\ v R x ) -> u = v ) /\ R e. Rels ) )
4	1 3	bitri	|- ( R e. Disjs <-> ( A. u A. v A. x ( ( u R x /\ v R x ) -> u = v ) /\ R e. Rels ) )