Definition df-wetr 4319

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 4505). 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 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))

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 4315	. 2 wff 𝑅 We 𝐴
4	1, 2	wfr 4313	. . 3 wff 𝑅 Fr 𝐴
5		vx	. . . . . . . . . 10 setvar 𝑥
6	5	cv 1347	. . . . . . . . 9 class 𝑥
7		vy	. . . . . . . . . 10 setvar 𝑦
8	7	cv 1347	. . . . . . . . 9 class 𝑦
9	6, 8, 2	wbr 3989	. . . . . . . 8 wff 𝑥𝑅𝑦
10		vz	. . . . . . . . . 10 setvar 𝑧
11	10	cv 1347	. . . . . . . . 9 class 𝑧
12	8, 11, 2	wbr 3989	. . . . . . . 8 wff 𝑦𝑅𝑧
13	9, 12	wa 103	. . . . . . 7 wff (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)
14	6, 11, 2	wbr 3989	. . . . . . 7 wff 𝑥𝑅𝑧
15	13, 14	wi 4	. . . . . 6 wff ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)
16	15, 10, 1	wral 2448	. . . . 5 wff ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)
17	16, 7, 1	wral 2448	. . . 4 wff ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)
18	17, 5, 1	wral 2448	. . 3 wff ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)
19	4, 18	wa 103	. 2 wff (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
20	3, 19	wb 104	1 wff (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))

This definition is referenced by:  nfwe  4340  weeq1  4341  weeq2  4342  wefr  4343  wepo  4344  wetrep  4345  we0  4346  ordwe  4560  wessep  4562  reg3exmidlemwe  4563