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Definition df-po 5319
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 27982). (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 5317 . 2 wff 𝑅 Po 𝐴
4 vx . . . . . . . . 9 setvar 𝑥
54cv 1506 . . . . . . . 8 class 𝑥
65, 5, 2wbr 4923 . . . . . . 7 wff 𝑥𝑅𝑥
76wn 3 . . . . . 6 wff ¬ 𝑥𝑅𝑥
8 vy . . . . . . . . . 10 setvar 𝑦
98cv 1506 . . . . . . . . 9 class 𝑦
105, 9, 2wbr 4923 . . . . . . . 8 wff 𝑥𝑅𝑦
11 vz . . . . . . . . . 10 setvar 𝑧
1211cv 1506 . . . . . . . . 9 class 𝑧
139, 12, 2wbr 4923 . . . . . . . 8 wff 𝑦𝑅𝑧
1410, 13wa 387 . . . . . . 7 wff (𝑥𝑅𝑦𝑦𝑅𝑧)
155, 12, 2wbr 4923 . . . . . . 7 wff 𝑥𝑅𝑧
1614, 15wi 4 . . . . . 6 wff ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
177, 16wa 387 . . . . 5 wff 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1817, 11, 1wral 3082 . . . 4 wff 𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1918, 8, 1wral 3082 . . 3 wff 𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
2019, 4, 1wral 3082 . 2 wff 𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
213, 20wb 198 1 wff (𝑅 Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
Colors of variables: wff setvar class
This definition is referenced by:  poss  5321  poeq1  5322  nfpo  5324  pocl  5326  ispod  5327  po0  5335  poinxp  5475  posn  5480  cnvpo  5970  isopolem  6915  porpss  7265  dfwe2  7306  poxp  7620  dfso3  32410  dfpo2  32451  elpotr  32486  poseq  32556
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