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Definition df-wetr 4256
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 4437). 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 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
Distinct variable groups:   𝑥,𝐴,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧

Detailed syntax breakdown of Definition df-wetr
StepHypRef Expression
1 cA . . 3 class 𝐴
2 cR . . 3 class 𝑅
31, 2wwe 4252 . 2 wff 𝑅 We 𝐴
41, 2wfr 4250 . . 3 wff 𝑅 Fr 𝐴
5 vx . . . . . . . . . 10 setvar 𝑥
65cv 1330 . . . . . . . . 9 class 𝑥
7 vy . . . . . . . . . 10 setvar 𝑦
87cv 1330 . . . . . . . . 9 class 𝑦
96, 8, 2wbr 3929 . . . . . . . 8 wff 𝑥𝑅𝑦
10 vz . . . . . . . . . 10 setvar 𝑧
1110cv 1330 . . . . . . . . 9 class 𝑧
128, 11, 2wbr 3929 . . . . . . . 8 wff 𝑦𝑅𝑧
139, 12wa 103 . . . . . . 7 wff (𝑥𝑅𝑦𝑦𝑅𝑧)
146, 11, 2wbr 3929 . . . . . . 7 wff 𝑥𝑅𝑧
1513, 14wi 4 . . . . . 6 wff ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
1615, 10, 1wral 2416 . . . . 5 wff 𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
1716, 7, 1wral 2416 . . . 4 wff 𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
1817, 5, 1wral 2416 . . 3 wff 𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
194, 18wa 103 . 2 wff (𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
203, 19wb 104 1 wff (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
Colors of variables: wff set class
This definition is referenced by:  nfwe  4277  weeq1  4278  weeq2  4279  wefr  4280  wepo  4281  wetrep  4282  we0  4283  ordwe  4490  wessep  4492  reg3exmidlemwe  4493
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