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Definition df-so 5469
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 10913). 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 5467 . 2 wff 𝑅 Or 𝐴
41, 2wpo 5466 . . 3 wff 𝑅 Po 𝐴
5 vx . . . . . . . 8 setvar 𝑥
65cv 1542 . . . . . . 7 class 𝑥
7 vy . . . . . . . 8 setvar 𝑦
87cv 1542 . . . . . . 7 class 𝑦
96, 8, 2wbr 5053 . . . . . 6 wff 𝑥𝑅𝑦
105, 7weq 1971 . . . . . 6 wff 𝑥 = 𝑦
118, 6, 2wbr 5053 . . . . . 6 wff 𝑦𝑅𝑥
129, 10, 11w3o 1088 . . . . 5 wff (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)
1312, 7, 1wral 3061 . . . 4 wff 𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)
1413, 5, 1wral 3061 . . 3 wff 𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)
154, 14wa 399 . 2 wff (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
163, 15wb 209 1 wff (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
Colors of variables: wff setvar class
This definition is referenced by:  nfso  5474  sopo  5487  soss  5488  soeq1  5489  solin  5493  issod  5501  so0  5504  soinxp  5630  sosn  5635  cnvso  6151  isosolem  7156  sorpss  7516  dfwe2  7559  soxp  7896  sornom  9891  zorn2lem6  10115  tosso  17925  dfso3  33379  dfso2  33440  soseq  33540
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