Definition df-po 5594

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 30368). (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 5592	. 2 wff 𝑅 Po 𝐴
4		vx	. . . . . . . . 9 setvar 𝑥
5	4	cv 1533	. . . . . . . 8 class 𝑥
6	5, 5, 2	wbr 5153	. . . . . . 7 wff 𝑥𝑅𝑥
7	6	wn 3	. . . . . 6 wff ¬ 𝑥𝑅𝑥
8		vy	. . . . . . . . . 10 setvar 𝑦
9	8	cv 1533	. . . . . . . . 9 class 𝑦
10	5, 9, 2	wbr 5153	. . . . . . . 8 wff 𝑥𝑅𝑦
11		vz	. . . . . . . . . 10 setvar 𝑧
12	11	cv 1533	. . . . . . . . 9 class 𝑧
13	9, 12, 2	wbr 5153	. . . . . . . 8 wff 𝑦𝑅𝑧
14	10, 13	wa 394	. . . . . . 7 wff (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)
15	5, 12, 2	wbr 5153	. . . . . . 7 wff 𝑥𝑅𝑧
16	14, 15	wi 4	. . . . . 6 wff ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)
17	7, 16	wa 394	. . . . 5 wff (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
18	17, 11, 1	wral 3051	. . . 4 wff ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
19	18, 8, 1	wral 3051	. . 3 wff ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
20	19, 4, 1	wral 3051	. 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  5596  poeq1  5597  nfpo  5599  pocl  5601  poclOLD  5602  ispod  5603  po0  5611  poinxp  5762  posn  5767  cnvpo  6298  dfpo2  6307  isopolem  7357  porpss  7738  dfwe2  7782  epweon  7783  poxp  8142  poseq  8172  dfso3  35542  elpotr  35605