Definition df-disjs 36742

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 36751).

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 36765. Alternate definitions are dfdisjs 36746, ... , dfdisjs5 36750. (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 36293	. 2 class Disjs
2		cdisjss 36292	. . 3 class Disjss
3		crels 36262	. . 3 class Rels
4	2, 3	cin 3882	. 2 class ( Disjss ∩ Rels )
5	1, 4	wceq 1539	1 wff Disjs = ( Disjss ∩ Rels )

This definition is referenced by:  dfdisjs  36746