Definition df-wetr 4460

Description: Define the well-ordering predicate. It is unusual to define "well-ordering" in the absence of excluded middle, but we mean an ordering which is like the ordering which we have for ordinals (for example, it does not entail trichotomy because ordinals do not have that as seen at ordtriexmid 4648). Given excluded middle, well-ordering is usually defined to require trichotomy (and the definition of Fr is typically also different). (Contributed by Mario Carneiro and Jim Kingdon, 23-Sep-2021.)

Assertion

Ref	Expression
df-wetr	|- ( R We A <-> ( R Fr A /\ A. x e. A A. y e. A A. z e. A ( ( x R y /\ y R z ) -> x R z ) ) )

Distinct variable groups:   x, A, y, z   x, R, y, z

Detailed syntax breakdown of Definition df-wetr

Step	Hyp	Ref	Expression
1		cA	. . 3 class A
2		cR	. . 3 class R
3	1, 2	wwe 4456	. 2 wff R We A
4	1, 2	wfr 4454	. . 3 wff R Fr A
5		vx	. . . . . . . . . 10 setvar x
6	5	cv 1397	. . . . . . . . 9 class x
7		vy	. . . . . . . . . 10 setvar y
8	7	cv 1397	. . . . . . . . 9 class y
9	6, 8, 2	wbr 4114	. . . . . . . 8 wff x R y
10		vz	. . . . . . . . . 10 setvar z
11	10	cv 1397	. . . . . . . . 9 class z
12	8, 11, 2	wbr 4114	. . . . . . . 8 wff y R z
13	9, 12	wa 104	. . . . . . 7 wff ( x R y /\ y R z )
14	6, 11, 2	wbr 4114	. . . . . . 7 wff x R z
15	13, 14	wi 4	. . . . . 6 wff ( ( x R y /\ y R z ) -> x R z )
16	15, 10, 1	wral 2522	. . . . 5 wff A. z e. A ( ( x R y /\ y R z ) -> x R z )
17	16, 7, 1	wral 2522	. . . 4 wff A. y e. A A. z e. A ( ( x R y /\ y R z ) -> x R z )
18	17, 5, 1	wral 2522	. . 3 wff A. x e. A A. y e. A A. z e. A ( ( x R y /\ y R z ) -> x R z )
19	4, 18	wa 104	. 2 wff ( R Fr A /\ A. x e. A A. y e. A A. z e. A ( ( x R y /\ y R z ) -> x R z ) )
20	3, 19	wb 105	1 wff ( R We A <-> ( R Fr A /\ A. x e. A A. y e. A A. z e. A ( ( x R y /\ y R z ) -> x R z ) ) )

This definition is referenced by:  nfwe  4481  weeq1  4482  weeq2  4483  wefr  4484  wepo  4485  wetrep  4486  we0  4487  ordwe  4703  wessep  4705  reg3exmidlemwe  4706