Theorem eldisjs

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 ) )

Proof

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 ) )