Definition df-iso 4312

Description: Define the strict linear order predicate. The expression 𝑅 Or 𝐴 is true if relationship 𝑅 orders 𝐴. The property 𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦) is called weak linearity by Proposition 11.2.3 of [HoTT], p. (varies). If we assumed excluded middle, it would be equivalent to trichotomy, 𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥. (Contributed by NM, 21-Jan-1996.) (Revised by Jim Kingdon, 4-Oct-2018.)

Assertion

Ref	Expression
df-iso	⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))

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

Detailed syntax breakdown of Definition df-iso

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cR	. . 3 class 𝑅
3	1, 2	wor 4310	. 2 wff 𝑅 Or 𝐴
4	1, 2	wpo 4309	. . 3 wff 𝑅 Po 𝐴
5		vx	. . . . . . . . 9 setvar 𝑥
6	5	cv 1363	. . . . . . . 8 class 𝑥
7		vy	. . . . . . . . 9 setvar 𝑦
8	7	cv 1363	. . . . . . . 8 class 𝑦
9	6, 8, 2	wbr 4018	. . . . . . 7 wff 𝑥𝑅𝑦
10		vz	. . . . . . . . . 10 setvar 𝑧
11	10	cv 1363	. . . . . . . . 9 class 𝑧
12	6, 11, 2	wbr 4018	. . . . . . . 8 wff 𝑥𝑅𝑧
13	11, 8, 2	wbr 4018	. . . . . . . 8 wff 𝑧𝑅𝑦
14	12, 13	wo 709	. . . . . . 7 wff (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)
15	9, 14	wi 4	. . . . . 6 wff (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))
16	15, 10, 1	wral 2468	. . . . 5 wff ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))
17	16, 7, 1	wral 2468	. . . 4 wff ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))
18	17, 5, 1	wral 2468	. . 3 wff ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))
19	4, 18	wa 104	. 2 wff (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
20	3, 19	wb 105	1 wff (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))

This definition is referenced by:  nfso  4317  sopo  4328  soss  4329  soeq1  4330  issod  4334  sowlin  4335  so0  4341  ordsoexmid  4576  soinxp  4711  sosng  4714  cnvsom  5187  isosolem  5841  ltsopr  7613  ltsosr  7781  ltso  8053  xrltso  9814