Description: The "Divide et Aequivalere" Theorem: every disjoint relation generates equivalent cosets by the relation: generalization of the former prter1 , cf. eldisjim . (Contributed by Peter Mazsa, 3-May-2019) (Revised by Peter Mazsa, 17-Sep-2021)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | disjim | |- ( Disj R -> EqvRel ,~ R ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dfdisjALTV4 | |- ( Disj R <-> ( A. y E* u u R y /\ Rel R ) ) |
|
| 2 | 1 | simplbi | |- ( Disj R -> A. y E* u u R y ) |
| 3 | trcoss | |- ( A. y E* u u R y -> A. x A. y A. z ( ( x ,~ R y /\ y ,~ R z ) -> x ,~ R z ) ) |
|
| 4 | 2 3 | syl | |- ( Disj R -> A. x A. y A. z ( ( x ,~ R y /\ y ,~ R z ) -> x ,~ R z ) ) |
| 5 | eqvrelcoss3 | |- ( EqvRel ,~ R <-> A. x A. y A. z ( ( x ,~ R y /\ y ,~ R z ) -> x ,~ R z ) ) |
|
| 6 | 4 5 | sylibr | |- ( Disj R -> EqvRel ,~ R ) |