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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 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.) |
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 4067 | . 2 wff 𝑅 We 𝐴 |
4 | 1, 2 | wfr 4065 | . . 3 wff 𝑅 Fr 𝐴 |
5 | vx | . . . . . . . . . 10 setvar 𝑥 | |
6 | 5 | cv 1242 | . . . . . . . . 9 class 𝑥 |
7 | vy | . . . . . . . . . 10 setvar 𝑦 | |
8 | 7 | cv 1242 | . . . . . . . . 9 class 𝑦 |
9 | 6, 8, 2 | wbr 3764 | . . . . . . . 8 wff 𝑥𝑅𝑦 |
10 | vz | . . . . . . . . . 10 setvar 𝑧 | |
11 | 10 | cv 1242 | . . . . . . . . 9 class 𝑧 |
12 | 8, 11, 2 | wbr 3764 | . . . . . . . 8 wff 𝑦𝑅𝑧 |
13 | 9, 12 | wa 97 | . . . . . . 7 wff (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) |
14 | 6, 11, 2 | wbr 3764 | . . . . . . 7 wff 𝑥𝑅𝑧 |
15 | 13, 14 | wi 4 | . . . . . 6 wff ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) |
16 | 15, 10, 1 | wral 2306 | . . . . 5 wff ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) |
17 | 16, 7, 1 | wral 2306 | . . . 4 wff ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) |
18 | 17, 5, 1 | wral 2306 | . . 3 wff ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) |
19 | 4, 18 | wa 97 | . 2 wff (𝑅 Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) |
20 | 3, 19 | wb 98 | 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 4092 weeq1 4093 weeq2 4094 wefr 4095 wepo 4096 wetrep 4097 we0 4098 ordwe 4300 wessep 4302 reg3exmidlemwe 4303 |
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