Definition df-disjs 38741

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 38750).

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 38764. Alternate definitions are dfdisjs 38745, ... , dfdisjs5 38749. (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 38247	. 2 class Disjs
2		cdisjss 38246	. . 3 class Disjss
3		crels 38216	. . 3 class Rels
4	2, 3	cin 3901	. 2 class ( Disjss ∩ Rels )
5	1, 4	wceq 1541	1 wff Disjs = ( Disjss ∩ Rels )

This definition is referenced by:  dfdisjs  38745