Theorem bnj1204 35026

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 1137	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) → 𝑅 FrSe 𝐴)
2		ssrab2 4080	. . . . . . 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 1139	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) → ∃𝑥 ∈ 𝐴 ¬ 𝜑)
5		rabn0 4389	. . . . . . 7 ⊢ ({𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ≠ ∅ ↔ ∃𝑥 ∈ 𝐴 ¬ 𝜑)
6	4, 5	sylibr 234	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) → {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ≠ ∅)
7		nfrab1 3457	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ¬ 𝜑}
8	7	nfcrii 2900	. . . . . . 7 ⊢ (𝑧 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} → ∀𝑥 𝑧 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑})
9	8	bnj1228 35025	. . . . . 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 1373	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) → ∃𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥)
11		biid 261	. . . . 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 1914	. . . . . . 7 ⊢ Ⅎ𝑥 𝑅 FrSe 𝐴
13		nfra1 3284	. . . . . . 7 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)
14		nfre1 3285	. . . . . . 7 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 ¬ 𝜑
15	12, 13, 14	nf3an 1901	. . . . . 6 ⊢ Ⅎ𝑥(𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑)
16	15	nf5ri 2195	. . . . 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 34865	. . . 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 2737	. . . . . 6 ⊢ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} = {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}
19	18, 11	bnj1212 34813	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑥 ∈ 𝐴)
20		nfra1 3284	. . . . . . . 8 ⊢ Ⅎ𝑦∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥
21		simp3 1139	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥)
22	21	bnj1211 34811	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑦(𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} → ¬ 𝑦𝑅𝑥))
23		con2b 359	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} → ¬ 𝑦𝑅𝑥) ↔ (𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}))
24	23	albii 1819	. . . . . . . . . . . . . 14 ⊢ (∀𝑦(𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} → ¬ 𝑦𝑅𝑥) ↔ ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}))
25	22, 24	sylib 218	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}))
26		simp2 1138	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑦𝑅𝑥)
27		sp 2183	. . . . . . . . . . . . 13 ⊢ (∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}) → (𝑦𝑅𝑥 → ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑}))
28	25, 26, 27	sylc 65	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → ¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑})
29		simp1 1137	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝑦 ∈ 𝐴)
30		nfcv 2905	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑥𝐴
31	30	elrabsf 3834	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ↔ (𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥] ¬ 𝜑))
32		vex 3484	. . . . . . . . . . . . . . . . . . 19 ⊢ 𝑦 ∈ V
33		sbcng 3836	. . . . . . . . . . . . . . . . . . 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 623	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝐴 ∧ [𝑦 / 𝑥] ¬ 𝜑) ↔ (𝑦 ∈ 𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
36	31, 35	bitri 275	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ↔ (𝑦 ∈ 𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
37	36	notbii 320	. . . . . . . . . . . . . . 15 ⊢ (¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ↔ ¬ (𝑦 ∈ 𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
38		imnan 399	. . . . . . . . . . . . . . 15 ⊢ ((𝑦 ∈ 𝐴 → ¬ ¬ [𝑦 / 𝑥]𝜑) ↔ ¬ (𝑦 ∈ 𝐴 ∧ ¬ [𝑦 / 𝑥]𝜑))
39	37, 38	sylbb2 238	. . . . . . . . . . . . . 14 ⊢ (¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} → (𝑦 ∈ 𝐴 → ¬ ¬ [𝑦 / 𝑥]𝜑))
40	39	imp 406	. . . . . . . . . . . . 13 ⊢ ((¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ 𝑦 ∈ 𝐴) → ¬ ¬ [𝑦 / 𝑥]𝜑)
41	40	notnotrd 133	. . . . . . . . . . . 12 ⊢ ((¬ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ 𝑦 ∈ 𝐴) → [𝑦 / 𝑥]𝜑)
42	28, 29, 41	syl2anc 584	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑)
43	42	3expa 1119	. . . . . . . . . 10 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥) ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑)
44	43	expcom 413	. . . . . . . . 9 ⊢ (∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → ((𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥) → [𝑦 / 𝑥]𝜑))
45	44	expd 415	. . . . . . . 8 ⊢ (∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → (𝑦 ∈ 𝐴 → (𝑦𝑅𝑥 → [𝑦 / 𝑥]𝜑)))
46	20, 45	ralrimi 3257	. . . . . . 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 234	. . . . . 6 ⊢ (∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥 → 𝜓)
49	48	3ad2ant3 1136	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝜓)
50		simp12 1205	. . . . 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 1139	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑))
52	51	bnj1211 34811	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → ∀𝑥(𝑥 ∈ 𝐴 → (𝜓 → 𝜑)))
53		simp1 1137	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → 𝑥 ∈ 𝐴)
54		simp2 1138	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → 𝜓)
55		sp 2183	. . . . . 6 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝜓 → 𝜑)) → (𝑥 ∈ 𝐴 → (𝜓 → 𝜑)))
56	52, 53, 54, 55	syl3c 66	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → 𝜑)
57	19, 49, 50, 56	syl3anc 1373	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ∧ ∀𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ¬ 𝑦𝑅𝑥) → 𝜑)
58		rabid 3458	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} ↔ (𝑥 ∈ 𝐴 ∧ ¬ 𝜑))
59	58	simprbi 496	. . . . 5 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ 𝜑} → ¬ 𝜑)
60	59	3ad2ant2 1135	. . . 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 34833	. . 3 ⊢ ¬ (𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑) ∧ ∃𝑥 ∈ 𝐴 ¬ 𝜑)
62	61	bnj1224 34815	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑)
63		dfral2 3099	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∃𝑥 ∈ 𝐴 ¬ 𝜑)
64	62, 63	sylibr 234	1 ⊢ ((𝑅 FrSe 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜓 → 𝜑)) → ∀𝑥 ∈ 𝐴 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087  ∀wal 1538   ∈ wcel 2108   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  {crab 3436  Vcvv 3480  [wsbc 3788   ⊆ wss 3951  ∅c0 4333   class class class wbr 5143   FrSe w-bnj15 34706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-reg 9632  ax-inf2 9681

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-om 7888  df-1o 8506  df-bnj17 34701  df-bnj14 34703  df-bnj13 34705  df-bnj15 34707  df-bnj18 34709  df-bnj19 34711

This theorem is referenced by:  bnj1417  35055