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Definition df-drs 18012
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 18010 . 2 class Dirset
2 vb . . . . . . . 8 setvar 𝑏
32cv 1538 . . . . . . 7 class 𝑏
4 c0 4258 . . . . . . 7 class
53, 4wne 2943 . . . . . 6 wff 𝑏 ≠ ∅
6 vx . . . . . . . . . . . 12 setvar 𝑥
76cv 1538 . . . . . . . . . . 11 class 𝑥
8 vz . . . . . . . . . . . 12 setvar 𝑧
98cv 1538 . . . . . . . . . . 11 class 𝑧
10 vr . . . . . . . . . . . 12 setvar 𝑟
1110cv 1538 . . . . . . . . . . 11 class 𝑟
127, 9, 11wbr 5076 . . . . . . . . . 10 wff 𝑥𝑟𝑧
13 vy . . . . . . . . . . . 12 setvar 𝑦
1413cv 1538 . . . . . . . . . . 11 class 𝑦
1514, 9, 11wbr 5076 . . . . . . . . . 10 wff 𝑦𝑟𝑧
1612, 15wa 396 . . . . . . . . 9 wff (𝑥𝑟𝑧𝑦𝑟𝑧)
1716, 8, 3wrex 3065 . . . . . . . 8 wff 𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧)
1817, 13, 3wral 3064 . . . . . . 7 wff 𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧)
1918, 6, 3wral 3064 . . . . . 6 wff 𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧)
205, 19wa 396 . . . . 5 wff (𝑏 ≠ ∅ ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧))
21 vf . . . . . . 7 setvar 𝑓
2221cv 1538 . . . . . 6 class 𝑓
23 cple 16967 . . . . . 6 class le
2422, 23cfv 6435 . . . . 5 class (le‘𝑓)
2520, 10, 24wsbc 3717 . . . 4 wff [(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧))
26 cbs 16910 . . . . 5 class Base
2722, 26cfv 6435 . . . 4 class (Base‘𝑓)
2825, 2, 27wsbc 3717 . . 3 wff [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧))
29 cproset 18009 . . 3 class Proset
3028, 21, 29crab 3068 . 2 class {𝑓 ∈ Proset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥𝑏𝑦𝑏𝑧𝑏 (𝑥𝑟𝑧𝑦𝑟𝑧))}
311, 30wceq 1539 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  18017
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