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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 4328 | . 2 wff 𝑅 Fr 𝐴 |
4 | vs | . . . . 5 setvar 𝑠 | |
5 | 4 | cv 1352 | . . . 4 class 𝑠 |
6 | 1, 2, 5 | wfrfor 4327 | . . 3 wff FrFor 𝑅𝐴𝑠 |
7 | 6, 4 | wal 1351 | . 2 wff ∀𝑠 FrFor 𝑅𝐴𝑠 |
8 | 3, 7 | wb 105 | 1 wff (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠) |
Colors of variables: wff set class |
This definition is referenced by: freq1 4344 freq2 4346 nffr 4349 frirrg 4350 fr0 4351 frind 4352 zfregfr 4573 |
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