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Definition df-drs 18341
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 18339 . 2 class Dirset
2 vb . . . . . . . 8 setvar 𝑏
32cv 1539 . . . . . . 7 class 𝑏
4 c0 4333 . . . . . . 7 class
53, 4wne 2940 . . . . . 6 wff 𝑏 ≠ ∅
6 vx . . . . . . . . . . . 12 setvar 𝑥
76cv 1539 . . . . . . . . . . 11 class 𝑥
8 vz . . . . . . . . . . . 12 setvar 𝑧
98cv 1539 . . . . . . . . . . 11 class 𝑧
10 vr . . . . . . . . . . . 12 setvar 𝑟
1110cv 1539 . . . . . . . . . . 11 class 𝑟
127, 9, 11wbr 5143 . . . . . . . . . 10 wff 𝑥𝑟𝑧
13 vy . . . . . . . . . . . 12 setvar 𝑦
1413cv 1539 . . . . . . . . . . 11 class 𝑦
1514, 9, 11wbr 5143 . . . . . . . . . 10 wff 𝑦𝑟𝑧
1612, 15wa 395 . . . . . . . . 9 wff (𝑥𝑟𝑧𝑦𝑟𝑧)
1716, 8, 3wrex 3070 . . . . . . . 8 wff 𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧)
1817, 13, 3wral 3061 . . . . . . 7 wff 𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧)
1918, 6, 3wral 3061 . . . . . 6 wff 𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧)
205, 19wa 395 . . . . 5 wff (𝑏 ≠ ∅ ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧))
21 vf . . . . . . 7 setvar 𝑓
2221cv 1539 . . . . . 6 class 𝑓
23 cple 17304 . . . . . 6 class le
2422, 23cfv 6561 . . . . 5 class (le‘𝑓)
2520, 10, 24wsbc 3788 . . . 4 wff [(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧))
26 cbs 17247 . . . . 5 class Base
2722, 26cfv 6561 . . . 4 class (Base‘𝑓)
2825, 2, 27wsbc 3788 . . 3 wff [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧))
29 cproset 18338 . . 3 class Proset
3028, 21, 29crab 3436 . 2 class {𝑓 ∈ Proset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧))}
311, 30wceq 1540 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  18347
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