Definition df-fr 5585

Description: Define the well-founded relation predicate. Definition 6.24(1) of [TakeutiZaring] p. 30. For alternate definitions, see dffr2 5593 and dffr3 6066. A class is called well-founded when the membership relation E (see df-eprel 5532) is well-founded on it, that is, 𝐴 is well-founded if E Fr 𝐴 (some sources request that the membership relation be well-founded on its transitive closure). (Contributed by NM, 3-Apr-1994.)

Assertion

Ref	Expression
df-fr	⊢ (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))

Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴,𝑦,𝑧

Detailed syntax breakdown of Definition df-fr

Step	Hyp	Ref	Expression
1		cA	. . 3 class 𝐴
2		cR	. . 3 class 𝑅
3	1, 2	wfr 5582	. 2 wff 𝑅 Fr 𝐴
4		vx	. . . . . . 7 setvar 𝑥
5	4	cv 1541	. . . . . 6 class 𝑥
6	5, 1	wss 3903	. . . . 5 wff 𝑥 ⊆ 𝐴
7		c0 4287	. . . . . 6 class ∅
8	5, 7	wne 2933	. . . . 5 wff 𝑥 ≠ ∅
9	6, 8	wa 395	. . . 4 wff (𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅)
10		vz	. . . . . . . . 9 setvar 𝑧
11	10	cv 1541	. . . . . . . 8 class 𝑧
12		vy	. . . . . . . . 9 setvar 𝑦
13	12	cv 1541	. . . . . . . 8 class 𝑦
14	11, 13, 2	wbr 5100	. . . . . . 7 wff 𝑧𝑅𝑦
15	14	wn 3	. . . . . 6 wff ¬ 𝑧𝑅𝑦
16	15, 10, 5	wral 3052	. . . . 5 wff ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦
17	16, 12, 5	wrex 3062	. . . 4 wff ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦
18	9, 17	wi 4	. . 3 wff ((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)
19	18, 4	wal 1540	. 2 wff ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦)
20	3, 19	wb 206	1 wff (𝑅 Fr 𝐴 ↔ ∀𝑥((𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅) → ∃𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦))

This definition is referenced by:  dffr6  5588  dffr2  5593  dffr2ALT  5594  frss  5596  freq1  5599  nffr  5605  frinxp  5715  frsn  5720  f1oweALT  7926  frxp  8078  frxp2  8096  frxp3  8103  frfi  9197  fpwwe2lem11  10564  fpwwe2lem12  10565  lrrecfr  27951  bnj1154  35174  vonf1owev  35321  dffr5  35967  dfon2lem9  36002  weiunfr  36680  finorwe  37626  fin2so  37847  fnwe2  43399