Definition df-disjs 39034

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 39043).

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 39069. Alternate definitions are dfdisjs 39038, ... , dfdisjs5 39042. (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 38463	. 2 class Disjs
2		cdisjss 38462	. . 3 class Disjss
3		crels 38430	. . 3 class Rels
4	2, 3	cin 3902	. 2 class ( Disjss ∩ Rels )
5	1, 4	wceq 1542	1 wff Disjs = ( Disjss ∩ Rels )

This definition is referenced by:  dfdisjs  39038  eldisjsim2  39180