MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-drs Structured version   Visualization version   GIF version

Definition df-drs 17397
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 ∈ Proset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧))}
Distinct variable group:   𝑓,𝑏,𝑟,𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-drs
StepHypRef Expression
1 cdrs 17395 . 2 class Dirset
2 vb . . . . . . . 8 setvar 𝑏
32cv 1506 . . . . . . 7 class 𝑏
4 c0 4179 . . . . . . 7 class
53, 4wne 2968 . . . . . 6 wff 𝑏 ≠ ∅
6 vx . . . . . . . . . . . 12 setvar 𝑥
76cv 1506 . . . . . . . . . . 11 class 𝑥
8 vz . . . . . . . . . . . 12 setvar 𝑧
98cv 1506 . . . . . . . . . . 11 class 𝑧
10 vr . . . . . . . . . . . 12 setvar 𝑟
1110cv 1506 . . . . . . . . . . 11 class 𝑟
127, 9, 11wbr 4929 . . . . . . . . . 10 wff 𝑥𝑟𝑧
13 vy . . . . . . . . . . . 12 setvar 𝑦
1413cv 1506 . . . . . . . . . . 11 class 𝑦
1514, 9, 11wbr 4929 . . . . . . . . . 10 wff 𝑦𝑟𝑧
1612, 15wa 387 . . . . . . . . 9 wff (𝑥𝑟𝑧𝑦𝑟𝑧)
1716, 8, 3wrex 3090 . . . . . . . 8 wff 𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧)
1817, 13, 3wral 3089 . . . . . . 7 wff 𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧)
1918, 6, 3wral 3089 . . . . . 6 wff 𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧)
205, 19wa 387 . . . . 5 wff (𝑏 ≠ ∅ ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧))
21 vf . . . . . . 7 setvar 𝑓
2221cv 1506 . . . . . 6 class 𝑓
23 cple 16428 . . . . . 6 class le
2422, 23cfv 6188 . . . . 5 class (le‘𝑓)
2520, 10, 24wsbc 3682 . . . 4 wff [(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧))
26 cbs 16339 . . . . 5 class Base
2722, 26cfv 6188 . . . 4 class (Base‘𝑓)
2825, 2, 27wsbc 3682 . . 3 wff [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧))
29 cproset 17394 . . 3 class Proset
3028, 21, 29crab 3093 . 2 class {𝑓 ∈ Proset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧))}
311, 30wceq 1507 1 wff Dirset = {𝑓 ∈ Proset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧))}
Colors of variables: wff setvar class
This definition is referenced by:  isdrs  17402
  Copyright terms: Public domain W3C validator