Definition df-drs 17929

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 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧))}

Distinct variable group:   𝑓,𝑏,𝑟,𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-drs

Step	Hyp	Ref	Expression
1		cdrs 17927	. 2 class Dirset
2		vb	. . . . . . . 8 setvar 𝑏
3	2	cv 1538	. . . . . . 7 class 𝑏
4		c0 4253	. . . . . . 7 class ∅
5	3, 4	wne 2942	. . . . . 6 wff 𝑏 ≠ ∅
6		vx	. . . . . . . . . . . 12 setvar 𝑥
7	6	cv 1538	. . . . . . . . . . 11 class 𝑥
8		vz	. . . . . . . . . . . 12 setvar 𝑧
9	8	cv 1538	. . . . . . . . . . 11 class 𝑧
10		vr	. . . . . . . . . . . 12 setvar 𝑟
11	10	cv 1538	. . . . . . . . . . 11 class 𝑟
12	7, 9, 11	wbr 5070	. . . . . . . . . 10 wff 𝑥𝑟𝑧
13		vy	. . . . . . . . . . . 12 setvar 𝑦
14	13	cv 1538	. . . . . . . . . . 11 class 𝑦
15	14, 9, 11	wbr 5070	. . . . . . . . . 10 wff 𝑦𝑟𝑧
16	12, 15	wa 395	. . . . . . . . 9 wff (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧)
17	16, 8, 3	wrex 3064	. . . . . . . 8 wff ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧)
18	17, 13, 3	wral 3063	. . . . . . 7 wff ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧)
19	18, 6, 3	wral 3063	. . . . . 6 wff ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧)
20	5, 19	wa 395	. . . . 5 wff (𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧))
21		vf	. . . . . . 7 setvar 𝑓
22	21	cv 1538	. . . . . 6 class 𝑓
23		cple 16895	. . . . . 6 class le
24	22, 23	cfv 6418	. . . . 5 class (le‘𝑓)
25	20, 10, 24	wsbc 3711	. . . 4 wff [(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧))
26		cbs 16840	. . . . 5 class Base
27	22, 26	cfv 6418	. . . 4 class (Base‘𝑓)
28	25, 2, 27	wsbc 3711	. . 3 wff [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧))
29		cproset 17926	. . 3 class Proset
30	28, 21, 29	crab 3067	. 2 class {𝑓 ∈ Proset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧))}
31	1, 30	wceq 1539	1 wff Dirset = {𝑓 ∈ Proset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧))}

This definition is referenced by:  isdrs  17934