Theorem bnj1204 35642

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 1154	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) → 𝑅 FrSe 𝐴)
2		ssrab2 4005	. . . . . . 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 1156	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) → ∃𝑥 ∈ 𝐴 ¬ 𝜑)
5		rabn0 4338	. . . . . . 7 ⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 ¬ 𝜑)
6	4, 5	sylibr 237	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) → {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ≠ ∅)
7		nfrab1 3431	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ¬ 𝜑}
8	7	nfcrii 2917	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} → ∀𝑥 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑})
9	8	bnj1228 35641	. . . . . 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 1398	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) → ∃𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥)
11		biid 264	. . . . 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 1946	. . . . . . 7 ⊢ Ⅎ𝑥 𝑅 FrSe 𝐴
13		nfra1 3286	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)
14		nfre1 3287	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 ¬ 𝜑
15	12, 13, 14	nf3an 1933	. . . . . 6 ⊢ Ⅎ𝑥(𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑)
16	15	nf5ri 2231	. . . . 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 35481	. . . 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 2760	. . . . . 6 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}
19	18, 11	bnj1212 35429	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑥 ∈ 𝐴)
20		nfra1 3286	. . . . . . . 8 ⊢ Ⅎ𝑦∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥
21		simp3 1156	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥)
22	21	bnj1211 35427	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑦(𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} → ¬ 𝑦𝑅𝑥))
23		con2b 362	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} → ¬ 𝑦𝑅𝑥) ↔ (𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}))
24	23	albii 1851	. . . . . . . . . . . . . 14 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} → ¬ 𝑦𝑅𝑥) ↔ ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}))
25	22, 24	sylib 221	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}))
26		simp2 1155	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑦𝑅𝑥)
27		sp 2219	. . . . . . . . . . . . 13 ⊢ (∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) → (𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}))
28	25, 26, 27	sylc 66	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑})
29		simp1 1154	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑦 ∈ 𝐴)
30		nfcv 2922	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥𝐴
31	30	elrabsf 3783	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ↔ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥] ¬ 𝜑))
32		vex 3454	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑦 ∈ V
33		sbcng 3785	. . . . . . . . . . . . . . . . . . 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 635	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥] ¬ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
36	31, 35	bitri 278	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ↔ (𝑦 ∈ 𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
37	36	notbii 323	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ↔ ¬ (𝑦 ∈ 𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
38		imnan 405	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐴 → ¬ ¬ [𝑦 / 𝑥]𝜑) ↔ ¬ (𝑦 ∈ 𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
39	37, 38	sylbb2 241	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} → (𝑦 ∈ 𝐴 → ¬ ¬ [𝑦 / 𝑥]𝜑))
40	39	imp 412	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ 𝑦 ∈ 𝐴) → ¬ ¬ [𝑦 / 𝑥]𝜑)
41	40	notnotrd 134	. . . . . . . . . . . 12 ⊢ ((¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ 𝑦 ∈ 𝐴) → [𝑦 / 𝑥]𝜑)
42	28, 29, 41	syl2anc 596	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑)
43	42	3expa 1136	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥) ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑)
44	43	expcom 419	. . . . . . . . 9 ⊢ (∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑))
45	44	expd 421	. . . . . . . 8 ⊢ (∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → (𝑦 ∈ 𝐴 → (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜑)))
46	20, 45	ralrimi 3260	. . . . . . 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 237	. . . . . 6 ⊢ (∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → 𝜓)
49	48	3ad2ant3 1153	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝜓)
50		simp12 1223	. . . . 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 1156	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑))
52	51	bnj1211 35427	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → ∀𝑥(𝑥 ∈ 𝐴 → (𝜓 → 𝜑)))
53		simp1 1154	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → 𝑥 ∈ 𝐴)
54		simp2 1155	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → 𝜓)
55		sp 2219	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝜓 → 𝜑)) → (𝑥 ∈ 𝐴 → (𝜓 → 𝜑)))
56	52, 53, 54, 55	syl3c 67	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → 𝜑)
57	19, 49, 50, 56	syl3anc 1398	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝜑)
58		rabid 3432	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑))
59	58	simprbi 503	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} → ¬ 𝜑)
60	59	3ad2ant2 1152	. . . 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 35449	. . 3 ⊢ ¬ (𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑)
62	61	bnj1224 35431	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑)
63		dfral2 3113	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑)
64	62, 63	sylibr 237	1 ⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → ∀𝑥 ∈ 𝐴 𝜑)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1567   ∈ wcel 2144   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450  [wsbc 3738   ⊆ wss 3895  ∅c0 4278   class class class wbr 5102   FrSe w-bnj15 35323

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-reg 9586  ax-inf2 9642

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-om 7878  df-1o 8476  df-bnj17 35318  df-bnj14 35320  df-bnj13 35322  df-bnj15 35324  df-bnj18 35326  df-bnj19 35328

This theorem is used by:  bnj1417  35671