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Metamath Proof Explorer

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 ) )