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Definition df-drs 18291
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 18289 . 2 class Dirset
2 vb . . . . . . . 8 setvar 𝑏
32cv 1532 . . . . . . 7 class 𝑏
4 c0 4322 . . . . . . 7 class
53, 4wne 2929 . . . . . 6 wff 𝑏 ≠ ∅
6 vx . . . . . . . . . . . 12 setvar 𝑥
76cv 1532 . . . . . . . . . . 11 class 𝑥
8 vz . . . . . . . . . . . 12 setvar 𝑧
98cv 1532 . . . . . . . . . . 11 class 𝑧
10 vr . . . . . . . . . . . 12 setvar 𝑟
1110cv 1532 . . . . . . . . . . 11 class 𝑟
127, 9, 11wbr 5149 . . . . . . . . . 10 wff 𝑥𝑟𝑧
13 vy . . . . . . . . . . . 12 setvar 𝑦
1413cv 1532 . . . . . . . . . . 11 class 𝑦
1514, 9, 11wbr 5149 . . . . . . . . . 10 wff 𝑦𝑟𝑧
1612, 15wa 394 . . . . . . . . 9 wff (𝑥𝑟𝑧𝑦𝑟𝑧)
1716, 8, 3wrex 3059 . . . . . . . 8 wff 𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧)
1817, 13, 3wral 3050 . . . . . . 7 wff 𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧)
1918, 6, 3wral 3050 . . . . . 6 wff 𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧)
205, 19wa 394 . . . . 5 wff (𝑏 ≠ ∅ ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧))
21 vf . . . . . . 7 setvar 𝑓
2221cv 1532 . . . . . 6 class 𝑓
23 cple 17243 . . . . . 6 class le
2422, 23cfv 6549 . . . . 5 class (le‘𝑓)
2520, 10, 24wsbc 3773 . . . 4 wff [(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧))
26 cbs 17183 . . . . 5 class Base
2722, 26cfv 6549 . . . 4 class (Base‘𝑓)
2825, 2, 27wsbc 3773 . . 3 wff [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧))
29 cproset 18288 . . 3 class Proset
3028, 21, 29crab 3418 . 2 class {𝑓 ∈ Proset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧))}
311, 30wceq 1533 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  18296
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