Definition df-fr 5637

Description: Define the well-founded relation predicate. Definition 6.24(1) of [TakeutiZaring] p. 30. For alternate definitions, see dffr2 5646 and dffr3 6117. A class is called well-founded when the membership relation E (see df-eprel 5584) 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 5634	. 2 wff 𝑅 Fr 𝐴
4		vx	. . . . . . 7 setvar 𝑥
5	4	cv 1539	. . . . . 6 class 𝑥
6	5, 1	wss 3951	. . . . 5 wff 𝑥 ⊆ 𝐴
7		c0 4333	. . . . . 6 class ∅
8	5, 7	wne 2940	. . . . 5 wff 𝑥 ≠ ∅
9	6, 8	wa 395	. . . 4 wff (𝑥 ⊆ 𝐴 ∧ 𝑥 ≠ ∅)
10		vz	. . . . . . . . 9 setvar 𝑧
11	10	cv 1539	. . . . . . . 8 class 𝑧
12		vy	. . . . . . . . 9 setvar 𝑦
13	12	cv 1539	. . . . . . . 8 class 𝑦
14	11, 13, 2	wbr 5143	. . . . . . 7 wff 𝑧𝑅𝑦
15	14	wn 3	. . . . . 6 wff ¬ 𝑧𝑅𝑦
16	15, 10, 5	wral 3061	. . . . 5 wff ∀𝑧 ∈ 𝑥 ¬ 𝑧𝑅𝑦
17	16, 12, 5	wrex 3070	. . . 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 1538	. 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  5640  friOLD  5643  dffr2  5646  dffr2ALT  5647  frss  5649  freq1  5652  nffr  5658  frinxp  5768  frsn  5773  f1oweALT  7997  frxp  8151  frxp2  8169  frxp3  8176  frfi  9321  fpwwe2lem11  10681  fpwwe2lem12  10682  lrrecfr  27976  bnj1154  35013  dffr5  35754  dfon2lem9  35792  weiunfr  36468  finorwe  37383  fin2so  37614  fnwe2  43065