Definition df-disjs 39545

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 39554).

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 39580. Alternate definitions are dfdisjs 39549, ... , dfdisjs5 39553. (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 38974	. 2 class Disjs
2		cdisjss 38973	. . 3 class Disjss
3		crels 38941	. . 3 class Rels
4	2, 3	cin 3901	. 2 class ( Disjss ∩ Rels )
5	1, 4	wceq 1570	1 wff Disjs = ( Disjss ∩ Rels )

This definition is used by:  dfdisjs  39549  eldisjsim2  39691