Definition df-wetr 4365

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 4553). 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 4361	. 2 wff R We A
4	1, 2	wfr 4359	. . 3 wff R Fr A
5		vx	. . . . . . . . . 10 setvar x
6	5	cv 1363	. . . . . . . . 9 class x
7		vy	. . . . . . . . . 10 setvar y
8	7	cv 1363	. . . . . . . . 9 class y
9	6, 8, 2	wbr 4029	. . . . . . . 8 wff x R y
10		vz	. . . . . . . . . 10 setvar z
11	10	cv 1363	. . . . . . . . 9 class z
12	8, 11, 2	wbr 4029	. . . . . . . 8 wff y R z
13	9, 12	wa 104	. . . . . . 7 wff ( x R y /\ y R z )
14	6, 11, 2	wbr 4029	. . . . . . 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 2472	. . . . 5 wff A. z e. A ( ( x R y /\ y R z ) -> x R z )
17	16, 7, 1	wral 2472	. . . 4 wff A. y e. A A. z e. A ( ( x R y /\ y R z ) -> x R z )
18	17, 5, 1	wral 2472	. . 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  4386  weeq1  4387  weeq2  4388  wefr  4389  wepo  4390  wetrep  4391  we0  4392  ordwe  4608  wessep  4610  reg3exmidlemwe  4611