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Definition df-so 5180
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 10302). 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 5178 . 2 wff 𝑅 Or 𝐴
41, 2wpo 5177 . . 3 wff 𝑅 Po 𝐴
5 vx . . . . . . . 8 setvar 𝑥
65cv 1623 . . . . . . 7 class 𝑥
7 vy . . . . . . . 8 setvar 𝑦
87cv 1623 . . . . . . 7 class 𝑦
96, 8, 2wbr 4796 . . . . . 6 wff 𝑥𝑅𝑦
105, 7weq 2032 . . . . . 6 wff 𝑥 = 𝑦
118, 6, 2wbr 4796 . . . . . 6 wff 𝑦𝑅𝑥
129, 10, 11w3o 1071 . . . . 5 wff (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)
1312, 7, 1wral 3042 . . . 4 wff 𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)
1413, 5, 1wral 3042 . . 3 wff 𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)
154, 14wa 383 . 2 wff (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
163, 15wb 196 1 wff (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
Colors of variables: wff setvar class
This definition is referenced by:  nfso  5185  sopo  5196  soss  5197  soeq1  5198  solin  5202  issod  5209  so0  5212  soinxp  5332  sosn  5337  cnvso  5827  isosolem  6752  sorpss  7099  dfwe2  7138  soxp  7450  sornom  9283  zorn2lem6  9507  tosso  17229  dfso3  31900  dfso2  31943  soseq  32052
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