Definition df-frind 4379

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 4375	. 2 wff R Fr A
4		vs	. . . . 5 setvar s
5	4	cv 1372	. . . 4 class s
6	1, 2, 5	wfrfor 4374	. . 3 wff FrFor R A s
7	6, 4	wal 1371	. 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  4391  freq2  4393  nffr  4396  frirrg  4397  fr0  4398  frind  4399  zfregfr  4622