Definition df-iso 4328

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

Assertion

Ref	Expression
df-iso	|- ( R Or A <-> ( R Po A /\ A. x e. A A. y e. A A. z e. A ( x R y -> ( x R z \/ z R y ) ) ) )

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 4326	. 2 wff R Or A
4	1, 2	wpo 4325	. . 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 4029	. . . . . . 7 wff x R y
10		vz	. . . . . . . . . 10 setvar z
11	10	cv 1363	. . . . . . . . 9 class z
12	6, 11, 2	wbr 4029	. . . . . . . 8 wff x R z
13	11, 8, 2	wbr 4029	. . . . . . . 8 wff z R y
14	12, 13	wo 709	. . . . . . 7 wff ( x R z \/ z R y )
15	9, 14	wi 4	. . . . . 6 wff ( x R y -> ( x R z \/ z R y ) )
16	15, 10, 1	wral 2472	. . . . 5 wff A. z e. A ( x R y -> ( x R z \/ z R y ) )
17	16, 7, 1	wral 2472	. . . 4 wff A. y e. A A. z e. A ( x R y -> ( x R z \/ z R y ) )
18	17, 5, 1	wral 2472	. . 3 wff A. x e. A A. y e. A A. z e. A ( x R y -> ( x R z \/ z R y ) )
19	4, 18	wa 104	. 2 wff ( R Po A /\ A. x e. A A. y e. A A. z e. A ( x R y -> ( x R z \/ z R y ) ) )
20	3, 19	wb 105	1 wff ( R Or A <-> ( R Po A /\ A. x e. A A. y e. A A. z e. A ( x R y -> ( x R z \/ z R y ) ) ) )

This definition is referenced by:  nfso  4333  sopo  4344  soss  4345  soeq1  4346  issod  4350  sowlin  4351  so0  4357  ordsoexmid  4594  soinxp  4729  sosng  4732  cnvsom  5209  isosolem  5867  ltsopr  7656  ltsosr  7824  ltso  8097  xrltso  9862