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Definition df-disjALTV 37570
Description: Define the disjoint relation predicate, i.e., the disjoint predicate. A disjoint relation is a converse function of the relation by dfdisjALTV 37578, see the comment of df-disjs 37569 why we need disjoint relations instead of converse functions anyway.

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 37592. Alternate definitions are dfdisjALTV 37578, ... , dfdisjALTV5 37582. (Contributed by Peter Mazsa, 17-Jul-2021.)

Assertion
Ref Expression
df-disjALTV ( Disj 𝑅 ↔ ( CnvRefRel ≀ 𝑅 ∧ Rel 𝑅))

Detailed syntax breakdown of Definition df-disjALTV
StepHypRef Expression
1 cR . . 3 class 𝑅
21wdisjALTV 37072 . 2 wff Disj 𝑅
31ccnv 5675 . . . . 5 class 𝑅
43ccoss 37038 . . . 4 class 𝑅
54wcnvrefrel 37047 . . 3 wff CnvRefRel ≀ 𝑅
61wrel 5681 . . 3 wff Rel 𝑅
75, 6wa 396 . 2 wff ( CnvRefRel ≀ 𝑅 ∧ Rel 𝑅)
82, 7wb 205 1 wff ( Disj 𝑅 ↔ ( CnvRefRel ≀ 𝑅 ∧ Rel 𝑅))
Colors of variables: wff setvar class
This definition is referenced by:  dfdisjALTV  37578  dfdisjALTV2  37579  eldisjsdisj  37592  disjrel  37595
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