Definition df-disjs 38113

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 38122).

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 38136. Alternate definitions are dfdisjs 38117, ... , dfdisjs5 38121. (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 37616	. 2 class Disjs
2		cdisjss 37615	. . 3 class Disjss
3		crels 37585	. . 3 class Rels
4	2, 3	cin 3943	. 2 class ( Disjss ∩ Rels )
5	1, 4	wceq 1534	1 wff Disjs = ( Disjss ∩ Rels )

This definition is referenced by:  dfdisjs  38117