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Definition df-so 5585
Description: Define the strict complete (linear) order predicate. The expression 𝑅 Or 𝐴 is true if relationship 𝑅 orders 𝐴. For example, < Or ℝ is true (ltso 11281). Equivalent to Definition 6.19(1) of [TakeutiZaring] p. 29. (Contributed by NM, 21-Jan-1996.)
Assertion
Ref Expression
df-so (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
Distinct variable groups:   𝑥,𝑦,𝑅   𝑥,𝐴,𝑦

Detailed syntax breakdown of Definition df-so
StepHypRef Expression
1 cA . . 3 class 𝐴
2 cR . . 3 class 𝑅
31, 2wor 5583 . 2 wff 𝑅 Or 𝐴
41, 2wpo 5582 . . 3 wff 𝑅 Po 𝐴
5 vx . . . . . . . 8 setvar 𝑥
65cv 1541 . . . . . . 7 class 𝑥
7 vy . . . . . . . 8 setvar 𝑦
87cv 1541 . . . . . . 7 class 𝑦
96, 8, 2wbr 5144 . . . . . 6 wff 𝑥𝑅𝑦
105, 7weq 1967 . . . . . 6 wff 𝑥 = 𝑦
118, 6, 2wbr 5144 . . . . . 6 wff 𝑦𝑅𝑥
129, 10, 11w3o 1087 . . . . 5 wff (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)
1312, 7, 1wral 3062 . . . 4 wff 𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)
1413, 5, 1wral 3062 . . 3 wff 𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)
154, 14wa 397 . 2 wff (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
163, 15wb 205 1 wff (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
Colors of variables: wff setvar class
This definition is referenced by:  nfso  5590  sopo  5603  soss  5604  soeq1  5605  solin  5609  issod  5617  so0  5620  soinxp  5752  sosn  5757  cnvso  6279  isosolem  7331  sorpss  7705  dfwe2  7748  epweon  7749  soxp  8102  soseq  8132  sornom  10259  zorn2lem6  10483  tosso  18359  dfso3  34620  dfso2  34656
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