Definition df-wetr 4370

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 4558). 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 4366	. 2 wff 𝑅 We 𝐴
4	1, 2	wfr 4364	. . 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 4034	. . . . . . . 8 wff 𝑥𝑅𝑦
10		vz	. . . . . . . . . 10 setvar 𝑧
11	10	cv 1363	. . . . . . . . 9 class 𝑧
12	8, 11, 2	wbr 4034	. . . . . . . 8 wff 𝑦𝑅𝑧
13	9, 12	wa 104	. . . . . . 7 wff (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧)
14	6, 11, 2	wbr 4034	. . . . . . 7 wff 𝑥𝑅𝑧
15	13, 14	wi 4	. . . . . 6 wff ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)
16	15, 10, 1	wral 2475	. . . . 5 wff ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)
17	16, 7, 1	wral 2475	. . . 4 wff ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)
18	17, 5, 1	wral 2475	. . 3 wff ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)
19	4, 18	wa 104	. 2 wff (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
20	3, 19	wb 105	1 wff (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))

This definition is referenced by:  nfwe  4391  weeq1  4392  weeq2  4393  wefr  4394  wepo  4395  wetrep  4396  we0  4397  ordwe  4613  wessep  4615  reg3exmidlemwe  4616