Definition df-iso 4219

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 4217	. 2 wff 𝑅 Or 𝐴
4	1, 2	wpo 4216	. . 3 wff 𝑅 Po 𝐴
5		vx	. . . . . . . . 9 setvar 𝑥
6	5	cv 1330	. . . . . . . 8 class 𝑥
7		vy	. . . . . . . . 9 setvar 𝑦
8	7	cv 1330	. . . . . . . 8 class 𝑦
9	6, 8, 2	wbr 3929	. . . . . . 7 wff 𝑥𝑅𝑦
10		vz	. . . . . . . . . 10 setvar 𝑧
11	10	cv 1330	. . . . . . . . 9 class 𝑧
12	6, 11, 2	wbr 3929	. . . . . . . 8 wff 𝑥𝑅𝑧
13	11, 8, 2	wbr 3929	. . . . . . . 8 wff 𝑧𝑅𝑦
14	12, 13	wo 697	. . . . . . 7 wff (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)
15	9, 14	wi 4	. . . . . 6 wff (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))
16	15, 10, 1	wral 2416	. . . . 5 wff ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))
17	16, 7, 1	wral 2416	. . . 4 wff ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))
18	17, 5, 1	wral 2416	. . 3 wff ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))
19	4, 18	wa 103	. 2 wff (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
20	3, 19	wb 104	1 wff (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))))

This definition is referenced by:  nfso  4224  sopo  4235  soss  4236  soeq1  4237  issod  4241  sowlin  4242  so0  4248  ordsoexmid  4477  soinxp  4609  sosng  4612  cnvsom  5082  isosolem  5725  ltsopr  7404  ltsosr  7572  ltso  7842  xrltso  9582