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Definition df-drs 18352
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 18350 . 2 class Dirset
2 vb . . . . . . . 8 setvar 𝑏
32cv 1535 . . . . . . 7 class 𝑏
4 c0 4338 . . . . . . 7 class
53, 4wne 2937 . . . . . 6 wff 𝑏 ≠ ∅
6 vx . . . . . . . . . . . 12 setvar 𝑥
76cv 1535 . . . . . . . . . . 11 class 𝑥
8 vz . . . . . . . . . . . 12 setvar 𝑧
98cv 1535 . . . . . . . . . . 11 class 𝑧
10 vr . . . . . . . . . . . 12 setvar 𝑟
1110cv 1535 . . . . . . . . . . 11 class 𝑟
127, 9, 11wbr 5147 . . . . . . . . . 10 wff 𝑥𝑟𝑧
13 vy . . . . . . . . . . . 12 setvar 𝑦
1413cv 1535 . . . . . . . . . . 11 class 𝑦
1514, 9, 11wbr 5147 . . . . . . . . . 10 wff 𝑦𝑟𝑧
1612, 15wa 395 . . . . . . . . 9 wff (𝑥𝑟𝑧𝑦𝑟𝑧)
1716, 8, 3wrex 3067 . . . . . . . 8 wff 𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧)
1817, 13, 3wral 3058 . . . . . . 7 wff 𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧)
1918, 6, 3wral 3058 . . . . . 6 wff 𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧)
205, 19wa 395 . . . . 5 wff (𝑏 ≠ ∅ ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧))
21 vf . . . . . . 7 setvar 𝑓
2221cv 1535 . . . . . 6 class 𝑓
23 cple 17304 . . . . . 6 class le
2422, 23cfv 6562 . . . . 5 class (le‘𝑓)
2520, 10, 24wsbc 3790 . . . 4 wff [(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧))
26 cbs 17244 . . . . 5 class Base
2722, 26cfv 6562 . . . 4 class (Base‘𝑓)
2825, 2, 27wsbc 3790 . . 3 wff [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧))
29 cproset 18349 . . 3 class Proset
3028, 21, 29crab 3432 . 2 class {𝑓 ∈ Proset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧))}
311, 30wceq 1536 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  18358
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