Definition df-so 5537

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 11223). 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 5535	. 2 wff 𝑅 Or 𝐴
4	1, 2	wpo 5534	. . 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 5086	. . . . . 6 wff 𝑥𝑅𝑦
10	5, 7	weq 1964	. . . . . 6 wff 𝑥 = 𝑦
11	8, 6, 2	wbr 5086	. . . . . 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  5543  sopo  5555  soss  5556  soeq1  5557  solin  5563  issod  5571  so0  5574  soinxp  5710  sosn  5715  cnvso  6250  isosolem  7299  sorpss  7679  dfwe2  7725  epweon  7726  soxp  8076  soseq  8106  sornom  10196  zorn2lem6  10420  tosso  18380  dfso3  35899  dfso2  35934