Metamath Proof Explorer

Theorem dfdisjs2

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

Ref Expression
Assertion dfdisjs2 
|- Disjs = { r e. Rels | ,~ `' r C_ _I }

Proof

Step Hyp Ref Expression
1 dfdisjs 
 |-  Disjs = { r e. Rels | ,~ `' r e. CnvRefRels }
2 cosscnvelrels 
 |-  ( r e. Rels -> ,~ `' r e. Rels )
3 2 biantrud 
 |-  ( r e. Rels -> ( ,~ `' r C_ _I <-> ( ,~ `' r C_ _I /\ ,~ `' r e. Rels ) ) )
4 cosselcnvrefrels2 
 |-  ( ,~ `' r e. CnvRefRels <-> ( ,~ `' r C_ _I /\ ,~ `' r e. Rels ) )
5 3 4 syl6rbbr 
 |-  ( r e. Rels -> ( ,~ `' r e. CnvRefRels <-> ,~ `' r C_ _I ) )
6 1 5 rabimbieq 
 |-  Disjs = { r e. Rels | ,~ `' r C_ _I }