Definition df-disjs 39110

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 39119).

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 39145. Alternate definitions are dfdisjs 39114, ... , dfdisjs5 39118. (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 38539	. 2 class Disjs
2		cdisjss 38538	. . 3 class Disjss
3		crels 38506	. . 3 class Rels
4	2, 3	cin 3888	. 2 class ( Disjss ∩ Rels )
5	1, 4	wceq 1542	1 wff Disjs = ( Disjss ∩ Rels )

This definition is referenced by:  dfdisjs  39114  eldisjsim2  39256