Definition df-disjs 38668

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 38677).

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 38691. Alternate definitions are dfdisjs 38672, ... , dfdisjs5 38676. (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 38178	. 2 class Disjs
2		cdisjss 38177	. . 3 class Disjss
3		crels 38147	. . 3 class Rels
4	2, 3	cin 3925	. 2 class ( Disjss ∩ Rels )
5	1, 4	wceq 1540	1 wff Disjs = ( Disjss ∩ Rels )

This definition is referenced by:  dfdisjs  38672