Definition df-po 5592

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 30454). (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 5590	. 2 wff 𝑅 Po 𝐴
4		vx	. . . . . . . . 9 setvar 𝑥
5	4	cv 1539	. . . . . . . 8 class 𝑥
6	5, 5, 2	wbr 5143	. . . . . . 7 wff 𝑥𝑅𝑥
7	6	wn 3	. . . . . 6 wff ¬ 𝑥𝑅𝑥
8		vy	. . . . . . . . . 10 setvar 𝑦
9	8	cv 1539	. . . . . . . . 9 class 𝑦
10	5, 9, 2	wbr 5143	. . . . . . . 8 wff 𝑥𝑅𝑦
11		vz	. . . . . . . . . 10 setvar 𝑧
12	11	cv 1539	. . . . . . . . 9 class 𝑧
13	9, 12, 2	wbr 5143	. . . . . . . 8 wff 𝑦𝑅𝑧
14	10, 13	wa 395	. . . . . . 7 wff (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)
15	5, 12, 2	wbr 5143	. . . . . . 7 wff 𝑥𝑅𝑧
16	14, 15	wi 4	. . . . . 6 wff ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)
17	7, 16	wa 395	. . . . 5 wff (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
18	17, 11, 1	wral 3061	. . . 4 wff ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
19	18, 8, 1	wral 3061	. . 3 wff ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
20	19, 4, 1	wral 3061	. 2 wff ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
21	3, 20	wb 206	1 wff (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))

This definition is referenced by:  poss  5594  poeq1  5595  nfpo  5598  pocl  5600  ispod  5601  po0  5609  poinxp  5766  posn  5771  cnvpo  6307  dfpo2  6316  isopolem  7365  porpss  7747  dfwe2  7794  epweon  7795  poxp  8153  poseq  8183  dfso3  35720  elpotr  35782  weiunpo  36466