Definition df-po 5589

Description: Define the strict partial order predicate. Definition of [Enderton] p. 168. The expression 𝑅 Po 𝐴 means 𝑅 is a partial order on 𝐴. For example, < Po ℝ is true, while ≤ Po ℝ is false (ex-po 29688). (Contributed by NM, 16-Mar-1997.)

Assertion

Ref	Expression
df-po	⊢ (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴,𝑦,𝑧

Detailed syntax breakdown of Definition df-po

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cR	. . 3 class 𝑅
3	1, 2	wpo 5587	. 2 wff 𝑅 Po 𝐴
4		vx	. . . . . . . . 9 setvar 𝑥
5	4	cv 1541	. . . . . . . 8 class 𝑥
6	5, 5, 2	wbr 5149	. . . . . . 7 wff 𝑥𝑅𝑥
7	6	wn 3	. . . . . 6 wff ¬ 𝑥𝑅𝑥
8		vy	. . . . . . . . . 10 setvar 𝑦
9	8	cv 1541	. . . . . . . . 9 class 𝑦
10	5, 9, 2	wbr 5149	. . . . . . . 8 wff 𝑥𝑅𝑦
11		vz	. . . . . . . . . 10 setvar 𝑧
12	11	cv 1541	. . . . . . . . 9 class 𝑧
13	9, 12, 2	wbr 5149	. . . . . . . 8 wff 𝑦𝑅𝑧
14	10, 13	wa 397	. . . . . . 7 wff (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)
15	5, 12, 2	wbr 5149	. . . . . . 7 wff 𝑥𝑅𝑧
16	14, 15	wi 4	. . . . . 6 wff ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)
17	7, 16	wa 397	. . . . 5 wff (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
18	17, 11, 1	wral 3062	. . . 4 wff ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
19	18, 8, 1	wral 3062	. . 3 wff ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
20	19, 4, 1	wral 3062	. 2 wff ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
21	3, 20	wb 205	1 wff (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))

This definition is referenced by:  poss  5591  poeq1  5592  nfpo  5594  pocl  5596  poclOLD  5597  ispod  5598  po0  5606  poinxp  5757  posn  5762  cnvpo  6287  dfpo2  6296  isopolem  7342  porpss  7717  dfwe2  7761  epweon  7762  poxp  8114  poseq  8144  dfso3  34689  elpotr  34753