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Definition df-wetr 4099
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 4275). 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 4095 . 2 wff 𝑅 We 𝐴
41, 2wfr 4093 . . 3 wff 𝑅 Fr 𝐴
5 vx . . . . . . . . . 10 setvar 𝑥
65cv 1258 . . . . . . . . 9 class 𝑥
7 vy . . . . . . . . . 10 setvar 𝑦
87cv 1258 . . . . . . . . 9 class 𝑦
96, 8, 2wbr 3792 . . . . . . . 8 wff 𝑥𝑅𝑦
10 vz . . . . . . . . . 10 setvar 𝑧
1110cv 1258 . . . . . . . . 9 class 𝑧
128, 11, 2wbr 3792 . . . . . . . 8 wff 𝑦𝑅𝑧
139, 12wa 101 . . . . . . 7 wff (𝑥𝑅𝑦𝑦𝑅𝑧)
146, 11, 2wbr 3792 . . . . . . 7 wff 𝑥𝑅𝑧
1513, 14wi 4 . . . . . 6 wff ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
1615, 10, 1wral 2323 . . . . 5 wff 𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
1716, 7, 1wral 2323 . . . 4 wff 𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
1817, 5, 1wral 2323 . . 3 wff 𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)
194, 18wa 101 . 2 wff (𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
203, 19wb 102 1 wff (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
Colors of variables: wff set class
This definition is referenced by:  nfwe  4120  weeq1  4121  weeq2  4122  wefr  4123  wepo  4124  wetrep  4125  we0  4126  ordwe  4328  wessep  4330  reg3exmidlemwe  4331
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