Definition df-po 4274

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. (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 4272	. 2 wff 𝑅 Po 𝐴
4		vx	. . . . . . . . 9 setvar 𝑥
5	4	cv 1342	. . . . . . . 8 class 𝑥
6	5, 5, 2	wbr 3982	. . . . . . 7 wff 𝑥𝑅𝑥
7	6	wn 3	. . . . . 6 wff ¬ 𝑥𝑅𝑥
8		vy	. . . . . . . . . 10 setvar 𝑦
9	8	cv 1342	. . . . . . . . 9 class 𝑦
10	5, 9, 2	wbr 3982	. . . . . . . 8 wff 𝑥𝑅𝑦
11		vz	. . . . . . . . . 10 setvar 𝑧
12	11	cv 1342	. . . . . . . . 9 class 𝑧
13	9, 12, 2	wbr 3982	. . . . . . . 8 wff 𝑦𝑅𝑧
14	10, 13	wa 103	. . . . . . 7 wff (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)
15	5, 12, 2	wbr 3982	. . . . . . 7 wff 𝑥𝑅𝑧
16	14, 15	wi 4	. . . . . 6 wff ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)
17	7, 16	wa 103	. . . . 5 wff (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
18	17, 11, 1	wral 2444	. . . 4 wff ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
19	18, 8, 1	wral 2444	. . 3 wff ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
20	19, 4, 1	wral 2444	. 2 wff ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
21	3, 20	wb 104	1 wff (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))

This definition is referenced by:  poss  4276  poeq1  4277  nfpo  4279  pocl  4281  ispod  4282  po0  4289  poinxp  4673  posng  4676  cnvpom  5146  isopolem  5790  poxp  6200