Definition df-wetr 4454

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 4642). 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 4450	. 2 wff R We A
4	1, 2	wfr 4448	. . 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 4108	. . . . . . . 8 wff x R y
10		vz	. . . . . . . . . 10 setvar z
11	10	cv 1397	. . . . . . . . 9 class z
12	8, 11, 2	wbr 4108	. . . . . . . 8 wff y R z
13	9, 12	wa 104	. . . . . . 7 wff ( x R y /\ y R z )
14	6, 11, 2	wbr 4108	. . . . . . 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 2520	. . . . 5 wff A. z e. A ( ( x R y /\ y R z ) -> x R z )
17	16, 7, 1	wral 2520	. . . 4 wff A. y e. A A. z e. A ( ( x R y /\ y R z ) -> x R z )
18	17, 5, 1	wral 2520	. . 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  4475  weeq1  4476  weeq2  4477  wefr  4478  wepo  4479  wetrep  4480  we0  4481  ordwe  4697  wessep  4699  reg3exmidlemwe  4700