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Definition df-frind 4332
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.)
Assertion
Ref Expression
df-frind (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠)
Distinct variable groups:   𝑅,𝑠   𝐴,𝑠

Detailed syntax breakdown of Definition df-frind
StepHypRef Expression
1 cA . . 3 class 𝐴
2 cR . . 3 class 𝑅
31, 2wfr 4328 . 2 wff 𝑅 Fr 𝐴
4 vs . . . . 5 setvar 𝑠
54cv 1352 . . . 4 class 𝑠
61, 2, 5wfrfor 4327 . . 3 wff FrFor 𝑅𝐴𝑠
76, 4wal 1351 . 2 wff 𝑠 FrFor 𝑅𝐴𝑠
83, 7wb 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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