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Mirrors > Home > ILE Home > Th. List > df-frind | GIF version |
Description: Define the well-founded relation predicate. In the presence of excluded middle, there are a variety of equivalent ways to define this. In our case, this definition, in terms of an inductive principle, works better than one along the lines of "there is an element which is minimal when A is ordered by R". Because 𝑠 is constrained to be a set (not a proper class) here, sometimes it may be necessary to use FrFor directly rather than via Fr. (Contributed by Jim Kingdon and Mario Carneiro, 21-Sep-2021.) |
Ref | Expression |
---|---|
df-frind | ⊢ (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cA | . . 3 class 𝐴 | |
2 | cR | . . 3 class 𝑅 | |
3 | 1, 2 | wfr 4258 | . 2 wff 𝑅 Fr 𝐴 |
4 | vs | . . . . 5 setvar 𝑠 | |
5 | 4 | cv 1331 | . . . 4 class 𝑠 |
6 | 1, 2, 5 | wfrfor 4257 | . . 3 wff FrFor 𝑅𝐴𝑠 |
7 | 6, 4 | wal 1330 | . 2 wff ∀𝑠 FrFor 𝑅𝐴𝑠 |
8 | 3, 7 | wb 104 | 1 wff (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠) |
Colors of variables: wff set class |
This definition is referenced by: freq1 4274 freq2 4276 nffr 4279 frirrg 4280 fr0 4281 frind 4282 zfregfr 4496 |
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