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Metamath Proof Explorer
Description: If the elements of A are disjoint, then it has equivalent coelements
(former prter1 ). Special case of disjim . (Contributed by Rodolfo
Medina, 13-Oct-2010) (Revised by Mario Carneiro, 12-Aug-2015) (Revised
by Peter Mazsa, 8-Feb-2018) ( Revised by Peter Mazsa, 23-Sep-2021.)
|
|
Ref |
Expression |
|
Assertion |
eldisjim |
|- ( ElDisj A -> CoElEqvRel A ) |
Proof
| Step |
Hyp |
Ref |
Expression |
| 1 |
|
disjim |
|- ( Disj ( `' _E |` A ) -> EqvRel ,~ ( `' _E |` A ) ) |
| 2 |
|
df-eldisj |
|- ( ElDisj A <-> Disj ( `' _E |` A ) ) |
| 3 |
|
df-coeleqvrel |
|- ( CoElEqvRel A <-> EqvRel ,~ ( `' _E |` A ) ) |
| 4 |
1 2 3
|
3imtr4i |
|- ( ElDisj A -> CoElEqvRel A ) |