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Definition df-wetr 4226
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 4407). 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 4222 . 2 wff 𝑅 We 𝐴
41, 2wfr 4220 . . 3 wff 𝑅 Fr 𝐴
5 vx . . . . . . . . . 10 setvar 𝑥
65cv 1315 . . . . . . . . 9 class 𝑥
7 vy . . . . . . . . . 10 setvar 𝑦
87cv 1315 . . . . . . . . 9 class 𝑦
96, 8, 2wbr 3899 . . . . . . . 8 wff 𝑥𝑅𝑦
10 vz . . . . . . . . . 10 setvar 𝑧
1110cv 1315 . . . . . . . . 9 class 𝑧
128, 11, 2wbr 3899 . . . . . . . 8 wff 𝑦𝑅𝑧
139, 12wa 103 . . . . . . 7 wff (𝑥𝑅𝑦𝑦𝑅𝑧)
146, 11, 2wbr 3899 . . . . . . 7 wff 𝑥𝑅𝑧
1513, 14wi 4 . . . . . 6 wff ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
1615, 10, 1wral 2393 . . . . 5 wff 𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
1716, 7, 1wral 2393 . . . 4 wff 𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
1817, 5, 1wral 2393 . . 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  4247  weeq1  4248  weeq2  4249  wefr  4250  wepo  4251  wetrep  4252  we0  4253  ordwe  4460  wessep  4462  reg3exmidlemwe  4463
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