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Definition df-po 5596
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 30463). (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 5594 . 2 wff 𝑅 Po 𝐴
4 vx . . . . . . . . 9 setvar 𝑥
54cv 1535 . . . . . . . 8 class 𝑥
65, 5, 2wbr 5147 . . . . . . 7 wff 𝑥𝑅𝑥
76wn 3 . . . . . 6 wff ¬ 𝑥𝑅𝑥
8 vy . . . . . . . . . 10 setvar 𝑦
98cv 1535 . . . . . . . . 9 class 𝑦
105, 9, 2wbr 5147 . . . . . . . 8 wff 𝑥𝑅𝑦
11 vz . . . . . . . . . 10 setvar 𝑧
1211cv 1535 . . . . . . . . 9 class 𝑧
139, 12, 2wbr 5147 . . . . . . . 8 wff 𝑦𝑅𝑧
1410, 13wa 395 . . . . . . 7 wff (𝑥𝑅𝑦𝑦𝑅𝑧)
155, 12, 2wbr 5147 . . . . . . 7 wff 𝑥𝑅𝑧
1614, 15wi 4 . . . . . 6 wff ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
177, 16wa 395 . . . . 5 wff 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1817, 11, 1wral 3058 . . . 4 wff 𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1918, 8, 1wral 3058 . . 3 wff 𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
2019, 4, 1wral 3058 . 2 wff 𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
213, 20wb 206 1 wff (𝑅 Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
Colors of variables: wff setvar class
This definition is referenced by:  poss  5598  poeq1  5599  nfpo  5602  pocl  5604  ispod  5605  po0  5613  poinxp  5768  posn  5773  cnvpo  6308  dfpo2  6317  isopolem  7364  porpss  7745  dfwe2  7792  epweon  7793  poxp  8151  poseq  8181  dfso3  35699  elpotr  35762  weiunpo  36447
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