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Definition df-wetr 4071
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 don't have that as seen at ordtriexmid 4247). Given excluded middle, well-ordering is usually defined to require trichotomy (and the defintion 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 4067 . 2 wff 𝑅 We 𝐴
41, 2wfr 4065 . . 3 wff 𝑅 Fr 𝐴
5 vx . . . . . . . . . 10 setvar 𝑥
65cv 1242 . . . . . . . . 9 class 𝑥
7 vy . . . . . . . . . 10 setvar 𝑦
87cv 1242 . . . . . . . . 9 class 𝑦
96, 8, 2wbr 3764 . . . . . . . 8 wff 𝑥𝑅𝑦
10 vz . . . . . . . . . 10 setvar 𝑧
1110cv 1242 . . . . . . . . 9 class 𝑧
128, 11, 2wbr 3764 . . . . . . . 8 wff 𝑦𝑅𝑧
139, 12wa 97 . . . . . . 7 wff (𝑥𝑅𝑦𝑦𝑅𝑧)
146, 11, 2wbr 3764 . . . . . . 7 wff 𝑥𝑅𝑧
1513, 14wi 4 . . . . . 6 wff ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
1615, 10, 1wral 2306 . . . . 5 wff 𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
1716, 7, 1wral 2306 . . . 4 wff 𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
1817, 5, 1wral 2306 . . 3 wff 𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
194, 18wa 97 . 2 wff (𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
203, 19wb 98 1 wff (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
Colors of variables: wff set class
This definition is referenced by:  nfwe  4092  weeq1  4093  weeq2  4094  wefr  4095  wepo  4096  wetrep  4097  we0  4098  ordwe  4300  wessep  4302  reg3exmidlemwe  4303
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