Theorem bnj1204 30334

Description: Well-founded induction. The proof has been taken from Chapter 4 of Don Monk's notes on Set Theory. See http://euclid.colorado.edu/~monkd/setth.pdf. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj1204.1	⊢ (𝜓 ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜑))

Assertion

Ref	Expression
bnj1204	⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → ∀𝑥 ∈ 𝐴 𝜑)

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

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥,𝑦)

Proof of Theorem bnj1204

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1054	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) → 𝑅 FrSe 𝐴)
2		ssrab2 3650	. . . . . . 7 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ⊆ 𝐴
3	2	a1i 11	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) → {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ⊆ 𝐴)
4		simp3 1056	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) → ∃𝑥 ∈ 𝐴 ¬ 𝜑)
5		rabn0 3912	. . . . . . 7 ⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 ¬ 𝜑)
6	4, 5	sylibr 223	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) → {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ≠ ∅)
7		nfrab1 3099	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ¬ 𝜑}
8	7	nfcrii 2744	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} → ∀𝑥 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑})
9	8	bnj1228 30333	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ⊆ 𝐴 ∧ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ≠ ∅) → ∃𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥)
10	1, 3, 6, 9	syl3anc 1318	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) → ∃𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥)
11		biid 250	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) ↔ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥))
12		nfv 1830	. . . . . . 7 ⊢ Ⅎ𝑥 𝑅 FrSe 𝐴
13		nfra1 2925	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)
14		nfre1 2988	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 ¬ 𝜑
15	12, 13, 14	nf3an 1819	. . . . . 6 ⊢ Ⅎ𝑥(𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑)
16	15	nf5ri 2053	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) → ∀𝑥(𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑))
17	10, 11, 16	bnj1521 30175	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) → ∃𝑥((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥))
18		eqid 2610	. . . . . 6 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}
19	18, 11	bnj1212 30124	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑥 ∈ 𝐴)
20		nfra1 2925	. . . . . . . 8 ⊢ Ⅎ𝑦∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥
21		simp3 1056	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥)
22	21	bnj1211 30122	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑦(𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} → ¬ 𝑦𝑅𝑥))
23		con2b 348	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} → ¬ 𝑦𝑅𝑥) ↔ (𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}))
24	23	albii 1737	. . . . . . . . . . . . . 14 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} → ¬ 𝑦𝑅𝑥) ↔ ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}))
25	22, 24	sylib 207	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}))
26		simp2 1055	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑦𝑅𝑥)
27		sp 2041	. . . . . . . . . . . . 13 ⊢ (∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) → (𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}))
28	25, 26, 27	sylc 63	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑})
29		simp1 1054	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑦 ∈ 𝐴)
30		nfcv 2751	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥𝐴
31	30	elrabsf 3441	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ↔ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥] ¬ 𝜑))
32		vex 3176	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑦 ∈ V
33		sbcng 3443	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ V → ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑))
34	32, 33	ax-mp 5	. . . . . . . . . . . . . . . . . 18 ⊢ ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
35	34	anbi2i 726	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥] ¬ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
36	31, 35	bitri 263	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ↔ (𝑦 ∈ 𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
37	36	notbii 309	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ↔ ¬ (𝑦 ∈ 𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
38		imnan 437	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐴 → ¬ ¬ [𝑦 / 𝑥]𝜑) ↔ ¬ (𝑦 ∈ 𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
39	37, 38	sylbb2 227	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} → (𝑦 ∈ 𝐴 → ¬ ¬ [𝑦 / 𝑥]𝜑))
40	39	imp 444	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ 𝑦 ∈ 𝐴) → ¬ ¬ [𝑦 / 𝑥]𝜑)
41	40	notnotrd 127	. . . . . . . . . . . 12 ⊢ ((¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ 𝑦 ∈ 𝐴) → [𝑦 / 𝑥]𝜑)
42	28, 29, 41	syl2anc 691	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑)
43	42	3expa 1257	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥) ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑)
44	43	expcom 450	. . . . . . . . 9 ⊢ (∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑))
45	44	expd 451	. . . . . . . 8 ⊢ (∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → (𝑦 ∈ 𝐴 → (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜑)))
46	20, 45	ralrimi 2940	. . . . . . 7 ⊢ (∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜑))
47		bnj1204.1	. . . . . . 7 ⊢ (𝜓 ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜑))
48	46, 47	sylibr 223	. . . . . 6 ⊢ (∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → 𝜓)
49	48	3ad2ant3 1077	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝜓)
50		simp12 1085	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑))
51		simp3 1056	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑))
52	51	bnj1211 30122	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → ∀𝑥(𝑥 ∈ 𝐴 → (𝜓 → 𝜑)))
53		simp1 1054	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → 𝑥 ∈ 𝐴)
54		simp2 1055	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → 𝜓)
55		sp 2041	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝜓 → 𝜑)) → (𝑥 ∈ 𝐴 → (𝜓 → 𝜑)))
56	52, 53, 54, 55	syl3c 64	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → 𝜑)
57	19, 49, 50, 56	syl3anc 1318	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝜑)
58		rabid 3095	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑))
59	58	simprbi 479	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} → ¬ 𝜑)
60	59	3ad2ant2 1076	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ¬ 𝜑)
61	17, 57, 60	bnj1304 30144	. . 3 ⊢ ¬ (𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑)
62	61	bnj1224 30126	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑)
63		dfral2 2977	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑)
64	62, 63	sylibr 223	1 ⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  Vcvv 3173  [wsbc 3402   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583   FrSe w-bnj15 30011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-bnj17 30006  df-bnj14 30008  df-bnj13 30010  df-bnj15 30012  df-bnj18 30014  df-bnj19 30016

This theorem is referenced by:  bnj1417  30363