Definition df-disjs 37877

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 37886).

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 37900. Alternate definitions are dfdisjs 37881, ... , dfdisjs5 37885. (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 37379	. 2 class Disjs
2		cdisjss 37378	. . 3 class Disjss
3		crels 37348	. . 3 class Rels
4	2, 3	cin 3946	. 2 class ( Disjss ∩ Rels )
5	1, 4	wceq 1539	1 wff Disjs = ( Disjss ∩ Rels )

This definition is referenced by:  dfdisjs  37881