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Definition df-so 4960
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 9997). 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 4958 . 2 wff 𝑅 Or 𝐴
41, 2wpo 4957 . . 3 wff 𝑅 Po 𝐴
5 vx . . . . . . . 8 setvar 𝑥
65cv 1474 . . . . . . 7 class 𝑥
7 vy . . . . . . . 8 setvar 𝑦
87cv 1474 . . . . . . 7 class 𝑦
96, 8, 2wbr 4583 . . . . . 6 wff 𝑥𝑅𝑦
105, 7weq 1861 . . . . . 6 wff 𝑥 = 𝑦
118, 6, 2wbr 4583 . . . . . 6 wff 𝑦𝑅𝑥
129, 10, 11w3o 1030 . . . . 5 wff (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)
1312, 7, 1wral 2896 . . . 4 wff 𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)
1413, 5, 1wral 2896 . . 3 wff 𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)
154, 14wa 383 . 2 wff (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
163, 15wb 195 1 wff (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
Colors of variables: wff setvar class
This definition is referenced by:  nfso  4965  sopo  4976  soss  4977  soeq1  4978  solin  4982  issod  4989  so0  4992  soinxp  5106  sosn  5111  cnvso  5591  isosolem  6497  sorpss  6840  dfwe2  6873  soxp  7177  sornom  8982  zorn2lem6  9206  tosso  16859  dfso3  30856  dfso2  30897  soseq  30995
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