Theorem dfdisjALTV3

Description: Alternate definition of the disjoint relation predicate, cf. dffunALTV3 . (Contributed by Peter Mazsa, 28-Jul-2021)

	Ref	Expression
Assertion	dfdisjALTV3	|- ( Disj R <-> ( A. u A. v A. x ( ( u R x /\ v R x ) -> u = v ) /\ Rel R ) )

Proof

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