Definition df-disjs 35977

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 35986).

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 36000. Alternate definitions are dfdisjs 35981, ... , dfdisjs5 35985. (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 35526	. 2 class Disjs
2		cdisjss 35525	. . 3 class Disjss
3		crels 35495	. . 3 class Rels
4	2, 3	cin 3928	. 2 class ( Disjss ∩ Rels )
5	1, 4	wceq 1536	1 wff Disjs = ( Disjss ∩ Rels )

This definition is referenced by:  dfdisjs  35981