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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 do not have that as seen at ordtriexmid 4521). 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.) |
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 4331 | . 2 wff 𝑅 We 𝐴 |
4 | 1, 2 | wfr 4329 | . . 3 wff 𝑅 Fr 𝐴 |
5 | vx | . . . . . . . . . 10 setvar 𝑥 | |
6 | 5 | cv 1352 | . . . . . . . . 9 class 𝑥 |
7 | vy | . . . . . . . . . 10 setvar 𝑦 | |
8 | 7 | cv 1352 | . . . . . . . . 9 class 𝑦 |
9 | 6, 8, 2 | wbr 4004 | . . . . . . . 8 wff 𝑥𝑅𝑦 |
10 | vz | . . . . . . . . . 10 setvar 𝑧 | |
11 | 10 | cv 1352 | . . . . . . . . 9 class 𝑧 |
12 | 8, 11, 2 | wbr 4004 | . . . . . . . 8 wff 𝑦𝑅𝑧 |
13 | 9, 12 | wa 104 | . . . . . . 7 wff (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) |
14 | 6, 11, 2 | wbr 4004 | . . . . . . 7 wff 𝑥𝑅𝑧 |
15 | 13, 14 | wi 4 | . . . . . 6 wff ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) |
16 | 15, 10, 1 | wral 2455 | . . . . 5 wff ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) |
17 | 16, 7, 1 | wral 2455 | . . . 4 wff ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) |
18 | 17, 5, 1 | wral 2455 | . . 3 wff ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) |
19 | 4, 18 | wa 104 | . 2 wff (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
20 | 3, 19 | wb 105 | 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 4356 weeq1 4357 weeq2 4358 wefr 4359 wepo 4360 wetrep 4361 we0 4362 ordwe 4576 wessep 4578 reg3exmidlemwe 4579 |
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