Definition df-iso 4333

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 4331	. 2 wff 𝑅 Or 𝐴
4	1, 2	wpo 4330	. . 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 4034	. . . . . . 7 wff 𝑥𝑅𝑦
10		vz	. . . . . . . . . 10 setvar 𝑧
11	10	cv 1363	. . . . . . . . 9 class 𝑧
12	6, 11, 2	wbr 4034	. . . . . . . 8 wff 𝑥𝑅𝑧
13	11, 8, 2	wbr 4034	. . . . . . . 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 2475	. . . . 5 wff ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))
17	16, 7, 1	wral 2475	. . . 4 wff ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))
18	17, 5, 1	wral 2475	. . 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  4338  sopo  4349  soss  4350  soeq1  4351  issod  4355  sowlin  4356  so0  4362  ordsoexmid  4599  soinxp  4734  sosng  4737  cnvsom  5214  isosolem  5874  ltsopr  7680  ltsosr  7848  ltso  8121  xrltso  9888