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Mirrors > Home > ILE Home > Th. List > df-wetr | GIF version |
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 4366). 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.) |
Ref | Expression |
---|---|
df-wetr | ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | cR | . . 3 class 𝑅 | |
3 | 1, 2 | wwe 4181 | . 2 wff 𝑅 We 𝐴 |
4 | 1, 2 | wfr 4179 | . . 3 wff 𝑅 Fr 𝐴 |
5 | vx | . . . . . . . . . 10 setvar 𝑥 | |
6 | 5 | cv 1295 | . . . . . . . . 9 class 𝑥 |
7 | vy | . . . . . . . . . 10 setvar 𝑦 | |
8 | 7 | cv 1295 | . . . . . . . . 9 class 𝑦 |
9 | 6, 8, 2 | wbr 3867 | . . . . . . . 8 wff 𝑥𝑅𝑦 |
10 | vz | . . . . . . . . . 10 setvar 𝑧 | |
11 | 10 | cv 1295 | . . . . . . . . 9 class 𝑧 |
12 | 8, 11, 2 | wbr 3867 | . . . . . . . 8 wff 𝑦𝑅𝑧 |
13 | 9, 12 | wa 103 | . . . . . . 7 wff (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) |
14 | 6, 11, 2 | wbr 3867 | . . . . . . 7 wff 𝑥𝑅𝑧 |
15 | 13, 14 | wi 4 | . . . . . 6 wff ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) |
16 | 15, 10, 1 | wral 2370 | . . . . 5 wff ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) |
17 | 16, 7, 1 | wral 2370 | . . . 4 wff ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) |
18 | 17, 5, 1 | wral 2370 | . . 3 wff ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) |
19 | 4, 18 | wa 103 | . 2 wff (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
20 | 3, 19 | wb 104 | 1 wff (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) |
Colors of variables: wff set class |
This definition is referenced by: nfwe 4206 weeq1 4207 weeq2 4208 wefr 4209 wepo 4210 wetrep 4211 we0 4212 ordwe 4419 wessep 4421 reg3exmidlemwe 4422 |
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