This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.
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 ) |