Definition df-disjs 35481

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 35490).

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 35504. Alternate definitions are dfdisjs 35485, ... , dfdisjs5 35489. (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 35031	. 2 class Disjs
2		cdisjss 35030	. . 3 class Disjss
3		crels 35000	. . 3 class Rels
4	2, 3	cin 3860	. 2 class ( Disjss ∩ Rels )
5	1, 4	wceq 1522	1 wff Disjs = ( Disjss ∩ Rels )

This definition is referenced by:  dfdisjs  35485