Definition df-disjs 39252

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 39261).

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 39287. Alternate definitions are dfdisjs 39256, ... , dfdisjs5 39260. (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 38681	. 2 class Disjs
2		cdisjss 38680	. . 3 class Disjss
3		crels 38648	. . 3 class Rels
4	2, 3	cin 3903	. 2 class ( Disjss ∩ Rels )
5	1, 4	wceq 1559	1 wff Disjs = ( Disjss ∩ Rels )

This definition is referenced by:  dfdisjs  39256  eldisjsim2  39398