Definition df-disjs 39689

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 39698).

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 39724. Alternate definitions are dfdisjs 39693, ... , dfdisjs5 39697. (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 39118	. 2 class Disjs
2		cdisjss 39117	. . 3 class Disjss
3		crels 39085	. . 3 class Rels
4	2, 3	cin 3898	. 2 class ( Disjss ∩ Rels )
5	1, 4	wceq 1570	1 wff Disjs = ( Disjss ∩ Rels )

This definition is used by:  dfdisjs  39693  eldisjsim2  39835