Definition df-disjs 38681

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 38690).

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 38704. Alternate definitions are dfdisjs 38685, ... , dfdisjs5 38689. (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 38187	. 2 class Disjs
2		cdisjss 38186	. . 3 class Disjss
3		crels 38156	. . 3 class Rels
4	2, 3	cin 3904	. 2 class ( Disjss ∩ Rels )
5	1, 4	wceq 1540	1 wff Disjs = ( Disjss ∩ Rels )

This definition is referenced by:  dfdisjs  38685