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Definition df-po 5005
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 27180). (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 5003 . 2 wff 𝑅 Po 𝐴
4 vx . . . . . . . . 9 setvar 𝑥
54cv 1479 . . . . . . . 8 class 𝑥
65, 5, 2wbr 4623 . . . . . . 7 wff 𝑥𝑅𝑥
76wn 3 . . . . . 6 wff ¬ 𝑥𝑅𝑥
8 vy . . . . . . . . . 10 setvar 𝑦
98cv 1479 . . . . . . . . 9 class 𝑦
105, 9, 2wbr 4623 . . . . . . . 8 wff 𝑥𝑅𝑦
11 vz . . . . . . . . . 10 setvar 𝑧
1211cv 1479 . . . . . . . . 9 class 𝑧
139, 12, 2wbr 4623 . . . . . . . 8 wff 𝑦𝑅𝑧
1410, 13wa 384 . . . . . . 7 wff (𝑥𝑅𝑦𝑦𝑅𝑧)
155, 12, 2wbr 4623 . . . . . . 7 wff 𝑥𝑅𝑧
1614, 15wi 4 . . . . . 6 wff ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
177, 16wa 384 . . . . 5 wff 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1817, 11, 1wral 2908 . . . 4 wff 𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
1918, 8, 1wral 2908 . . 3 wff 𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
2019, 4, 1wral 2908 . 2 wff 𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
213, 20wb 196 1 wff (𝑅 Po 𝐴 ↔ ∀𝑥𝐴𝑦𝐴𝑧𝐴𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
Colors of variables: wff setvar class
This definition is referenced by:  poss  5007  poeq1  5008  nfpo  5010  pocl  5012  ispod  5013  po0  5020  poinxp  5153  posn  5158  cnvpo  5642  isopolem  6560  porpss  6906  dfwe2  6943  poxp  7249  dfso3  31363  dfpo2  31406  elpotr  31440  poseq  31504
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