Definition df-so 5559

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 11390). 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 5557	. 2 wff 𝑅 Or 𝐴
4	1, 2	wpo 5556	. . 3 wff 𝑅 Po 𝐴
5		vx	. . . . . . . 8 setvar 𝑥
6	5	cv 1568	. . . . . . 7 class 𝑥
7		vy	. . . . . . . 8 setvar 𝑦
8	7	cv 1568	. . . . . . 7 class 𝑦
9	6, 8, 2	wbr 5102	. . . . . 6 wff 𝑥𝑅𝑦
10	5, 7	weq 1994	. . . . . 6 wff 𝑥 = 𝑦
11	8, 6, 2	wbr 5102	. . . . . 6 wff 𝑦𝑅𝑥
12	9, 10, 11	w3o 1102	. . . . 5 wff (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)
13	12, 7, 1	wral 3076	. . . 4 wff ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)
14	13, 5, 1	wral 3076	. . 3 wff ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)
15	4, 14	wa 401	. 2 wff (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
16	3, 15	wb 209	1 wff (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))

This definition is used by:  nfso  5565  sopo  5577  soss  5578  soeq1  5579  solin  5585  issod  5593  so0  5596  soinxp  5732  sosn  5737  cnvso  6291  isosolem  7355  sorpss  7744  dfwe2  7788  epweon  7789  soxp  8141  soseq  8176  sornom  10355  zorn2lem6  10579  tosso  18591  dfso3  36527  dfso2  36562