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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 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴,𝑦,𝑧

Detailed syntax breakdown of Definition df-po
StepHypRef Expression
1 cA . . 3 class 𝐴
2 cR . . 3 class 𝑅
31, 2wpo 5592 . 2 wff 𝑅 Po 𝐴
4 vx . . . . . . . . 9 setvar 𝑥
54cv 1533 . . . . . . . 8 class 𝑥
65, 5, 2wbr 5153 . . . . . . 7 wff 𝑥𝑅𝑥
76wn 3 . . . . . 6 wff ¬ 𝑥𝑅𝑥
8 vy . . . . . . . . . 10 setvar 𝑦
98cv 1533 . . . . . . . . 9 class 𝑦
105, 9, 2wbr 5153 . . . . . . . 8 wff 𝑥𝑅𝑦
11 vz . . . . . . . . . 10 setvar 𝑧
1211cv 1533 . . . . . . . . 9 class 𝑧
139, 12, 2wbr 5153 . . . . . . . 8 wff 𝑦𝑅𝑧
1410, 13wa 394 . . . . . . 7 wff (𝑥𝑅𝑦𝑦𝑅𝑧)
155, 12, 2wbr 5153 . . . . . . 7 wff 𝑥𝑅𝑧
1614, 15wi 4 . . . . . 6 wff ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
177, 16wa 394 . . . . 5 wff 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1817, 11, 1wral 3051 . . . 4 wff 𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1918, 8, 1wral 3051 . . 3 wff 𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
2019, 4, 1wral 3051 . 2 wff 𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
213, 20wb 205 1 wff (𝑅 Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
Colors of variables: wff setvar class
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
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