Definition df-so 5543

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 11227). 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 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))

Distinct variable groups:   𝑥,𝑦,𝑅   𝑥,𝐴,𝑦

Detailed syntax breakdown of Definition df-so

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cR	. . 3 class 𝑅
3	1, 2	wor 5541	. 2 wff 𝑅 Or 𝐴
4	1, 2	wpo 5540	. . 3 wff 𝑅 Po 𝐴
5		vx	. . . . . . . 8 setvar 𝑥
6	5	cv 1541	. . . . . . 7 class 𝑥
7		vy	. . . . . . . 8 setvar 𝑦
8	7	cv 1541	. . . . . . 7 class 𝑦
9	6, 8, 2	wbr 5100	. . . . . 6 wff 𝑥𝑅𝑦
10	5, 7	weq 1964	. . . . . 6 wff 𝑥 = 𝑦
11	8, 6, 2	wbr 5100	. . . . . 6 wff 𝑦𝑅𝑥
12	9, 10, 11	w3o 1086	. . . . 5 wff (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)
13	12, 7, 1	wral 3052	. . . 4 wff ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)
14	13, 5, 1	wral 3052	. . 3 wff ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)
15	4, 14	wa 395	. 2 wff (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
16	3, 15	wb 206	1 wff (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))

This definition is referenced by:  nfso  5549  sopo  5561  soss  5562  soeq1  5563  solin  5569  issod  5577  so0  5580  soinxp  5716  sosn  5721  cnvso  6256  isosolem  7305  sorpss  7685  dfwe2  7731  epweon  7732  soxp  8083  soseq  8113  sornom  10201  zorn2lem6  10425  tosso  18354  dfso3  35942  dfso2  35977