Definition df-iso 4400

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 4398	. 2 wff R Or A
4	1, 2	wpo 4397	. . 3 wff R Po A
5		vx	. . . . . . . . 9 setvar x
6	5	cv 1397	. . . . . . . 8 class x
7		vy	. . . . . . . . 9 setvar y
8	7	cv 1397	. . . . . . . 8 class y
9	6, 8, 2	wbr 4093	. . . . . . 7 wff x R y
10		vz	. . . . . . . . . 10 setvar z
11	10	cv 1397	. . . . . . . . 9 class z
12	6, 11, 2	wbr 4093	. . . . . . . 8 wff x R z
13	11, 8, 2	wbr 4093	. . . . . . . 8 wff z R y
14	12, 13	wo 716	. . . . . . 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 2511	. . . . 5 wff A. z e. A ( x R y -> ( x R z \/ z R y ) )
17	16, 7, 1	wral 2511	. . . 4 wff A. y e. A A. z e. A ( x R y -> ( x R z \/ z R y ) )
18	17, 5, 1	wral 2511	. . 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  4405  sopo  4416  soss  4417  soeq1  4418  issod  4422  sowlin  4423  so0  4429  ordsoexmid  4666  soinxp  4802  sosng  4805  cnvsom  5287  isosolem  5975  ltsopr  7876  ltsosr  8044  ltso  8316  xrltso  10092