Definition df-wetr 4429

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 4617). 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 4425	. 2 wff R We A
4	1, 2	wfr 4423	. . 3 wff R Fr A
5		vx	. . . . . . . . . 10 setvar x
6	5	cv 1394	. . . . . . . . 9 class x
7		vy	. . . . . . . . . 10 setvar y
8	7	cv 1394	. . . . . . . . 9 class y
9	6, 8, 2	wbr 4086	. . . . . . . 8 wff x R y
10		vz	. . . . . . . . . 10 setvar z
11	10	cv 1394	. . . . . . . . 9 class z
12	8, 11, 2	wbr 4086	. . . . . . . 8 wff y R z
13	9, 12	wa 104	. . . . . . 7 wff ( x R y /\ y R z )
14	6, 11, 2	wbr 4086	. . . . . . 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 2508	. . . . 5 wff A. z e. A ( ( x R y /\ y R z ) -> x R z )
17	16, 7, 1	wral 2508	. . . 4 wff A. y e. A A. z e. A ( ( x R y /\ y R z ) -> x R z )
18	17, 5, 1	wral 2508	. . 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  4450  weeq1  4451  weeq2  4452  wefr  4453  wepo  4454  wetrep  4455  we0  4456  ordwe  4672  wessep  4674  reg3exmidlemwe  4675