Definition df-drs 18305

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 18303	. 2 class Dirset
2		vb	. . . . . . . 8 setvar 𝑏
3	2	cv 1539	. . . . . . 7 class 𝑏
4		c0 4308	. . . . . . 7 class ∅
5	3, 4	wne 2932	. . . . . 6 wff 𝑏 ≠ ∅
6		vx	. . . . . . . . . . . 12 setvar 𝑥
7	6	cv 1539	. . . . . . . . . . 11 class 𝑥
8		vz	. . . . . . . . . . . 12 setvar 𝑧
9	8	cv 1539	. . . . . . . . . . 11 class 𝑧
10		vr	. . . . . . . . . . . 12 setvar 𝑟
11	10	cv 1539	. . . . . . . . . . 11 class 𝑟
12	7, 9, 11	wbr 5119	. . . . . . . . . 10 wff 𝑥𝑟𝑧
13		vy	. . . . . . . . . . . 12 setvar 𝑦
14	13	cv 1539	. . . . . . . . . . 11 class 𝑦
15	14, 9, 11	wbr 5119	. . . . . . . . . 10 wff 𝑦𝑟𝑧
16	12, 15	wa 395	. . . . . . . . 9 wff (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧)
17	16, 8, 3	wrex 3060	. . . . . . . 8 wff ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧)
18	17, 13, 3	wral 3051	. . . . . . 7 wff ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧)
19	18, 6, 3	wral 3051	. . . . . 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 1539	. . . . . 6 class 𝑓
23		cple 17276	. . . . . 6 class le
24	22, 23	cfv 6530	. . . . 5 class (le‘𝑓)
25	20, 10, 24	wsbc 3765	. . . 4 wff [(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧))
26		cbs 17226	. . . . 5 class Base
27	22, 26	cfv 6530	. . . 4 class (Base‘𝑓)
28	25, 2, 27	wsbc 3765	. . 3 wff [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧))
29		cproset 18302	. . 3 class Proset
30	28, 21, 29	crab 3415	. 2 class {𝑓 ∈ Proset ∣ [(Base‘𝑓) / 𝑏][(le‘𝑓) / 𝑟](𝑏 ≠ ∅ ∧ ∀𝑥 ∈ 𝑏 ∀𝑦 ∈ 𝑏 ∃𝑧 ∈ 𝑏 (𝑥𝑟𝑧 ∧ 𝑦𝑟𝑧))}
31	1, 30	wceq 1540	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  18311