Theorem dfdisjALTV5a

Description: Alternate definition of the disjoint relation predicate. Disj R means: different domain generators have disjoint cosets (unless the generators are equal), plus Rel R for relation-typedness. This is the characterization that makes canonicity/uniqueness arguments modular. It is the starting point for the entire " Disj <-> unique representative per block" pipeline that feeds into Disjs , see dfdisjs7 . (Contributed by Peter Mazsa, 3-Feb-2026)

	Ref	Expression
Assertion	dfdisjALTV5a	|- ( Disj R <-> ( A. u e. dom R A. v e. dom R ( ( [ u ] R i^i [ v ] R ) =/= (/) -> u = v ) /\ Rel R ) )

Proof

Step	Hyp	Ref	Expression
1		dfdisjALTV5	|- ( Disj R <-> ( A. u e. dom R A. v e. dom R ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) /\ Rel R ) )
2		orcom	|- ( ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) <-> ( ( [ u ] R i^i [ v ] R ) = (/) \/ u = v ) )
3		neor	|- ( ( ( [ u ] R i^i [ v ] R ) = (/) \/ u = v ) <-> ( ( [ u ] R i^i [ v ] R ) =/= (/) -> u = v ) )
4	2 3	bitri	|- ( ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) <-> ( ( [ u ] R i^i [ v ] R ) =/= (/) -> u = v ) )
5	4	2ralbii	|- ( A. u e. dom R A. v e. dom R ( u = v \/ ( [ u ] R i^i [ v ] R ) = (/) ) <-> A. u e. dom R A. v e. dom R ( ( [ u ] R i^i [ v ] R ) =/= (/) -> u = v ) )
6	1 5	bianbi	|- ( Disj R <-> ( A. u e. dom R A. v e. dom R ( ( [ u ] R i^i [ v ] R ) =/= (/) -> u = v ) /\ Rel R ) )