Definition df-po 5567

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 30923). (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 5565	. 2 wff 𝑅 Po 𝐴
4		vx	. . . . . . . . 9 setvar 𝑥
5	4	cv 1569	. . . . . . . 8 class 𝑥
6	5, 5, 2	wbr 5107	. . . . . . 7 wff 𝑥𝑅𝑥
7	6	wn 3	. . . . . 6 wff ¬ 𝑥𝑅𝑥
8		vy	. . . . . . . . . 10 setvar 𝑦
9	8	cv 1569	. . . . . . . . 9 class 𝑦
10	5, 9, 2	wbr 5107	. . . . . . . 8 wff 𝑥𝑅𝑦
11		vz	. . . . . . . . . 10 setvar 𝑧
12	11	cv 1569	. . . . . . . . 9 class 𝑧
13	9, 12, 2	wbr 5107	. . . . . . . 8 wff 𝑦𝑅𝑧
14	10, 13	wa 401	. . . . . . 7 wff (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)
15	5, 12, 2	wbr 5107	. . . . . . 7 wff 𝑥𝑅𝑧
16	14, 15	wi 4	. . . . . 6 wff ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)
17	7, 16	wa 401	. . . . 5 wff (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
18	17, 11, 1	wral 3078	. . . 4 wff ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
19	18, 8, 1	wral 3078	. . 3 wff ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
20	19, 4, 1	wral 3078	. 2 wff ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
21	3, 20	wb 209	1 wff (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))

This definition is used by:  poss  5569  poeq1  5570  nfpo  5573  pocl  5575  ispod  5576  po0  5584  poinxp  5740  posn  5745  cnvpo  6289  dfpo2  6298  isopolem  7350  porpss  7732  dfwe2  7777  epweon  7778  poxp  8130  poseq  8160  dfso3  36307  elpotr  36366  weiunpo  37092