Definition df-so 5593

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 11341). 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 5591	. 2 wff 𝑅 Or 𝐴
4	1, 2	wpo 5590	. . 3 wff 𝑅 Po 𝐴
5		vx	. . . . . . . 8 setvar 𝑥
6	5	cv 1539	. . . . . . 7 class 𝑥
7		vy	. . . . . . . 8 setvar 𝑦
8	7	cv 1539	. . . . . . 7 class 𝑦
9	6, 8, 2	wbr 5143	. . . . . 6 wff 𝑥𝑅𝑦
10	5, 7	weq 1962	. . . . . 6 wff 𝑥 = 𝑦
11	8, 6, 2	wbr 5143	. . . . . 6 wff 𝑦𝑅𝑥
12	9, 10, 11	w3o 1086	. . . . 5 wff (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)
13	12, 7, 1	wral 3061	. . . 4 wff ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)
14	13, 5, 1	wral 3061	. . 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  5599  sopo  5611  soss  5612  soeq1  5613  solin  5619  issod  5627  so0  5630  soinxp  5767  sosn  5772  cnvso  6308  isosolem  7367  sorpss  7748  dfwe2  7794  epweon  7795  soxp  8154  soseq  8184  sornom  10317  zorn2lem6  10541  tosso  18464  dfso3  35720  dfso2  35755