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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 38694).
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 38708. Alternate definitions are dfdisjs 38689, ... , dfdisjs5 38693. (Contributed by Peter Mazsa, 17-Jul-2021.) |
Ref | Expression |
---|---|
df-disjs | ⊢ Disjs = ( Disjss ∩ Rels ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdisjs 38194 | . 2 class Disjs | |
2 | cdisjss 38193 | . . 3 class Disjss | |
3 | crels 38163 | . . 3 class Rels | |
4 | 2, 3 | cin 3961 | . 2 class ( Disjss ∩ Rels ) |
5 | 1, 4 | wceq 1536 | 1 wff Disjs = ( Disjss ∩ Rels ) |
Colors of variables: wff setvar class |
This definition is referenced by: dfdisjs 38689 |
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