Definition df-so 5590

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 11294). 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 5588	. 2 wff 𝑅 Or 𝐴
4	1, 2	wpo 5587	. . 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 5149	. . . . . 6 wff 𝑥𝑅𝑦
10	5, 7	weq 1967	. . . . . 6 wff 𝑥 = 𝑦
11	8, 6, 2	wbr 5149	. . . . . 6 wff 𝑦𝑅𝑥
12	9, 10, 11	w3o 1087	. . . . 5 wff (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)
13	12, 7, 1	wral 3062	. . . 4 wff ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)
14	13, 5, 1	wral 3062	. . 3 wff ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)
15	4, 14	wa 397	. 2 wff (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
16	3, 15	wb 205	1 wff (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))

This definition is referenced by:  nfso  5595  sopo  5608  soss  5609  soeq1  5610  solin  5614  issod  5622  so0  5625  soinxp  5758  sosn  5763  cnvso  6288  isosolem  7344  sorpss  7718  dfwe2  7761  epweon  7762  soxp  8115  soseq  8145  sornom  10272  zorn2lem6  10496  tosso  18372  dfso3  34689  dfso2  34725