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Definition df-frind 4097
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 4093 . 2 wff 𝑅 Fr 𝐴
4 vs . . . . 5 setvar 𝑠
54cv 1258 . . . 4 class 𝑠
61, 2, 5wfrfor 4092 . . 3 wff FrFor 𝑅𝐴𝑠
76, 4wal 1257 . 2 wff 𝑠 FrFor 𝑅𝐴𝑠
83, 7wb 102 1 wff (𝑅 Fr 𝐴 ↔ ∀𝑠 FrFor 𝑅𝐴𝑠)
Colors of variables: wff set class
This definition is referenced by:  freq1  4109  freq2  4111  nffr  4114  frirrg  4115  fr0  4116  frind  4117  zfregfr  4326
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