Definition df-iso 4081

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 4079	. 2 wff R Or A
4	1, 2	wpo 4078	. . 3 wff R Po A
5		vx	. . . . . . . . 9 setvar x
6	5	cv 1284	. . . . . . . 8 class x
7		vy	. . . . . . . . 9 setvar y
8	7	cv 1284	. . . . . . . 8 class y
9	6, 8, 2	wbr 3806	. . . . . . 7 wff x R y
10		vz	. . . . . . . . . 10 setvar z
11	10	cv 1284	. . . . . . . . 9 class z
12	6, 11, 2	wbr 3806	. . . . . . . 8 wff x R z
13	11, 8, 2	wbr 3806	. . . . . . . 8 wff z R y
14	12, 13	wo 662	. . . . . . 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 2353	. . . . 5 wff A. z e. A ( x R y -> ( x R z \/ z R y ) )
17	16, 7, 1	wral 2353	. . . 4 wff A. y e. A A. z e. A ( x R y -> ( x R z \/ z R y ) )
18	17, 5, 1	wral 2353	. . 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 102	. 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 103	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  4086  sopo  4097  soss  4098  soeq1  4099  issod  4103  sowlin  4104  so0  4110  ordsoexmid  4334  soinxp  4457  sosng  4460  cnvsom  4912  isosolem  5516  ltsopr  6925  ltsosr  7080  ltso  7333  xrltso  9024