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Definition df-drs 15419
Description: Define the class of directed sets. A directed set is a nonempty preordered set where every pair of elements have some upper bound. Note that it is not required that there exist a least upper bound.

There is no consensus in the literature over whether directed sets are allowed to be empty. It is slightly more convenient for us if they are not. (Contributed by Stefan O'Rear, 1-Feb-2015.)

Assertion
Ref Expression
df-drs  |- Dirset  =  {
f  e.  Preset  |  [. ( Base `  f )  /  b ]. [. ( le `  f )  / 
r ]. ( b  =/=  (/)  /\  A. x  e.  b  A. y  e.  b  E. z  e.  b  ( x r z  /\  y r z ) ) }
Distinct variable group:    f, b, r, x, y, z

Detailed syntax breakdown of Definition df-drs
StepHypRef Expression
1 cdrs 15417 . 2  class Dirset
2 vb . . . . . . . 8  setvar  b
32cv 1378 . . . . . . 7  class  b
4 c0 3785 . . . . . . 7  class  (/)
53, 4wne 2662 . . . . . 6  wff  b  =/=  (/)
6 vx . . . . . . . . . . . 12  setvar  x
76cv 1378 . . . . . . . . . . 11  class  x
8 vz . . . . . . . . . . . 12  setvar  z
98cv 1378 . . . . . . . . . . 11  class  z
10 vr . . . . . . . . . . . 12  setvar  r
1110cv 1378 . . . . . . . . . . 11  class  r
127, 9, 11wbr 4447 . . . . . . . . . 10  wff  x r z
13 vy . . . . . . . . . . . 12  setvar  y
1413cv 1378 . . . . . . . . . . 11  class  y
1514, 9, 11wbr 4447 . . . . . . . . . 10  wff  y r z
1612, 15wa 369 . . . . . . . . 9  wff  ( x r z  /\  y
r z )
1716, 8, 3wrex 2815 . . . . . . . 8  wff  E. z  e.  b  ( x
r z  /\  y
r z )
1817, 13, 3wral 2814 . . . . . . 7  wff  A. y  e.  b  E. z  e.  b  ( x
r z  /\  y
r z )
1918, 6, 3wral 2814 . . . . . 6  wff  A. x  e.  b  A. y  e.  b  E. z  e.  b  ( x
r z  /\  y
r z )
205, 19wa 369 . . . . 5  wff  ( b  =/=  (/)  /\  A. x  e.  b  A. y  e.  b  E. z  e.  b  ( x
r z  /\  y
r z ) )
21 vf . . . . . . 7  setvar  f
2221cv 1378 . . . . . 6  class  f
23 cple 14565 . . . . . 6  class  le
2422, 23cfv 5588 . . . . 5  class  ( le
`  f )
2520, 10, 24wsbc 3331 . . . 4  wff  [. ( le `  f )  / 
r ]. ( b  =/=  (/)  /\  A. x  e.  b  A. y  e.  b  E. z  e.  b  ( x r z  /\  y r z ) )
26 cbs 14493 . . . . 5  class  Base
2722, 26cfv 5588 . . . 4  class  ( Base `  f )
2825, 2, 27wsbc 3331 . . 3  wff  [. ( Base `  f )  / 
b ]. [. ( le
`  f )  / 
r ]. ( b  =/=  (/)  /\  A. x  e.  b  A. y  e.  b  E. z  e.  b  ( x r z  /\  y r z ) )
29 cpreset 15416 . . 3  class  Preset
3028, 21, 29crab 2818 . 2  class  { f  e.  Preset  |  [. ( Base `  f )  / 
b ]. [. ( le
`  f )  / 
r ]. ( b  =/=  (/)  /\  A. x  e.  b  A. y  e.  b  E. z  e.  b  ( x r z  /\  y r z ) ) }
311, 30wceq 1379 1  wff Dirset  =  {
f  e.  Preset  |  [. ( Base `  f )  /  b ]. [. ( le `  f )  / 
r ]. ( b  =/=  (/)  /\  A. x  e.  b  A. y  e.  b  E. z  e.  b  ( x r z  /\  y r z ) ) }
Colors of variables: wff setvar class
This definition is referenced by:  isdrs  15424
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