| Mathbox for Peter Mazsa |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > df-disjs | Structured version Visualization version GIF version | ||
| Description: Define the disjoint
relations class, i.e., the class of disjoints. We
need Disjs for the definition of Parts and Part
for the
Partition-Equivalence Theorems: this need for Parts as disjoint relations
on their domain quotients is the reason why we must define Disjs
instead of simply using converse functions (cf. dfdisjALTV 38714).
The element of the class of disjoints and the disjoint predicate are the same, that is (𝑅 ∈ Disjs ↔ Disj 𝑅) when 𝑅 is a set, see eldisjsdisj 38728. Alternate definitions are dfdisjs 38709, ... , dfdisjs5 38713. (Contributed by Peter Mazsa, 17-Jul-2021.) |
| Ref | Expression |
|---|---|
| df-disjs | ⊢ Disjs = ( Disjss ∩ Rels ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cdisjs 38215 | . 2 class Disjs | |
| 2 | cdisjss 38214 | . . 3 class Disjss | |
| 3 | crels 38184 | . . 3 class Rels | |
| 4 | 2, 3 | cin 3950 | . 2 class ( Disjss ∩ Rels ) |
| 5 | 1, 4 | wceq 1540 | 1 wff Disjs = ( Disjss ∩ Rels ) |
| Colors of variables: wff setvar class |
| This definition is referenced by: dfdisjs 38709 |
| Copyright terms: Public domain | W3C validator |