Definition df-disjs 36815

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 36824).

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 36838. Alternate definitions are dfdisjs 36819, ... , dfdisjs5 36823. (Contributed by Peter Mazsa, 17-Jul-2021.)

Assertion

Ref	Expression
df-disjs	⊢ Disjs = ( Disjss ∩ Rels )

Detailed syntax breakdown of Definition df-disjs

Step	Hyp	Ref	Expression
1		cdisjs 36366	. 2 class Disjs
2		cdisjss 36365	. . 3 class Disjss
3		crels 36335	. . 3 class Rels
4	2, 3	cin 3886	. 2 class ( Disjss ∩ Rels )
5	1, 4	wceq 1539	1 wff Disjs = ( Disjss ∩ Rels )

This definition is referenced by:  dfdisjs  36819