Mathbox for Peter Mazsa < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  df-disjALTV Structured version   Visualization version   GIF version

Definition df-disjALTV 36138
 Description: Define the disjoint relation predicate, i.e., the disjoint predicate. A disjoint relation is a converse function of the relation by dfdisjALTV 36146, see the comment of df-disjs 36137 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 36160. Alternate definitions are dfdisjALTV 36146, ... , dfdisjALTV5 36150. (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 35687 . 2 wff Disj 𝑅
31ccnv 5519 . . . . 5 class 𝑅
43ccoss 35653 . . . 4 class 𝑅
54wcnvrefrel 35662 . . 3 wff CnvRefRel ≀ 𝑅
61wrel 5525 . . 3 wff Rel 𝑅
75, 6wa 399 . 2 wff ( CnvRefRel ≀ 𝑅 ∧ Rel 𝑅)
82, 7wb 209 1 wff ( Disj 𝑅 ↔ ( CnvRefRel ≀ 𝑅 ∧ Rel 𝑅))
 Colors of variables: wff setvar class This definition is referenced by:  dfdisjALTV  36146  dfdisjALTV2  36147  eldisjsdisj  36160  disjrel  36163
 Copyright terms: Public domain W3C validator