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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 37886).
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 37900. Alternate definitions are dfdisjs 37881, ... , dfdisjs5 37885. (Contributed by Peter Mazsa, 17-Jul-2021.) |
Ref | Expression |
---|---|
df-disjs | ⊢ Disjs = ( Disjss ∩ Rels ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdisjs 37379 | . 2 class Disjs | |
2 | cdisjss 37378 | . . 3 class Disjss | |
3 | crels 37348 | . . 3 class Rels | |
4 | 2, 3 | cin 3947 | . 2 class ( Disjss ∩ Rels ) |
5 | 1, 4 | wceq 1541 | 1 wff Disjs = ( Disjss ∩ Rels ) |
Colors of variables: wff setvar class |
This definition is referenced by: dfdisjs 37881 |
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