Definition df-disjs 37563

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 37572).

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 37586. Alternate definitions are dfdisjs 37567, ... , dfdisjs5 37571. (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 37065	. 2 class Disjs
2		cdisjss 37064	. . 3 class Disjss
3		crels 37034	. . 3 class Rels
4	2, 3	cin 3947	. 2 class ( Disjss ∩ Rels )
5	1, 4	wceq 1542	1 wff Disjs = ( Disjss ∩ Rels )

This definition is referenced by:  dfdisjs  37567