Metamath Proof Explorer

Theorem dfdisjALTV4

Description: Alternate definition of the disjoint relation predicate, cf. dffunALTV4 . (Contributed by Peter Mazsa, 5-Sep-2021)

Ref Expression
Assertion dfdisjALTV4 
|- ( Disj R <-> ( A. x E* u u R x /\ Rel R ) )

Proof

Step Hyp Ref Expression
1 dfdisjALTV2 
 |-  ( Disj R <-> ( ,~ `' R C_ _I /\ Rel R ) )
2 cosscnvssid4 
 |-  ( ,~ `' R C_ _I <-> A. x E* u u R x )
3 2 anbi1i 
 |-  ( ( ,~ `' R C_ _I /\ Rel R ) <-> ( A. x E* u u R x /\ Rel R ) )
4 1 3 bitri 
 |-  ( Disj R <-> ( A. x E* u u R x /\ Rel R ) )