Definition df-wetr 4481

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 4670). 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 4477	. 2 wff R We A
4	1, 2	wfr 4475	. . 3 wff R Fr A
5		vx	. . . . . . . . . 10 setvar x
6	5	cv 1401	. . . . . . . . 9 class x
7		vy	. . . . . . . . . 10 setvar y
8	7	cv 1401	. . . . . . . . 9 class y
9	6, 8, 2	wbr 4131	. . . . . . . 8 wff x R y
10		vz	. . . . . . . . . 10 setvar z
11	10	cv 1401	. . . . . . . . 9 class z
12	8, 11, 2	wbr 4131	. . . . . . . 8 wff y R z
13	9, 12	wa 104	. . . . . . 7 wff ( x R y /\ y R z )
14	6, 11, 2	wbr 4131	. . . . . . 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 2528	. . . . 5 wff A. z e. A ( ( x R y /\ y R z ) -> x R z )
17	16, 7, 1	wral 2528	. . . 4 wff A. y e. A A. z e. A ( ( x R y /\ y R z ) -> x R z )
18	17, 5, 1	wral 2528	. . 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 used by:  nfwe  4502  weeq1  4503  weeq2  4504  wefr  4505  wepo  4506  wetrep  4507  we0  4508  ordwe  4725  wessep  4727  reg3exmidlemwe  4728