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Definition df-frind 4317
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  <->  A. sFrFor  R A s )
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 4313 . 2  wff  R  Fr  A
4 vs . . . . 5  setvar  s
54cv 1347 . . . 4  class  s
61, 2, 5wfrfor 4312 . . 3  wff FrFor  R A s
76, 4wal 1346 . 2  wff  A. sFrFor  R A s
83, 7wb 104 1  wff  ( R  Fr  A  <->  A. sFrFor  R A s )
Colors of variables: wff set class
This definition is referenced by:  freq1  4329  freq2  4331  nffr  4334  frirrg  4335  fr0  4336  frind  4337  zfregfr  4558
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