Definition df-frind 4367

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

Step	Hyp	Ref	Expression
1		cA	. . 3 class A
2		cR	. . 3 class R
3	1, 2	wfr 4363	. 2 wff R Fr A
4		vs	. . . . 5 setvar s
5	4	cv 1363	. . . 4 class s
6	1, 2, 5	wfrfor 4362	. . 3 wff FrFor R A s
7	6, 4	wal 1362	. 2 wff A. sFrFor R A s
8	3, 7	wb 105	1 wff ( R Fr A <-> A. sFrFor R A s )

This definition is referenced by:  freq1  4379  freq2  4381  nffr  4384  frirrg  4385  fr0  4386  frind  4387  zfregfr  4610