Theorem exfzdc 10175

Description: Decidability of the existence of an integer defined by a decidable proposition. (Contributed by Jim Kingdon, 28-Jan-2022.)

Hypotheses

Ref	Expression
exfzdc.1	⊢ (𝜑 → 𝑀 ∈ ℤ)
exfzdc.2	⊢ (𝜑 → 𝑁 ∈ ℤ)
exfzdc.3	⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑁)) → DECID 𝜓)

Assertion

Ref	Expression
exfzdc	⊢ (𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑁)𝜓)

Distinct variable groups:   𝑛,𝑀   𝑛,𝑁   𝜑,𝑛

Allowed substitution hint:   𝜓(𝑛)

Proof of Theorem exfzdc

Dummy variables 𝑤 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		exfzdc.1	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
2		exfzdc.2	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℤ)
3		eluz 9479	. . . . 5 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁))
4	1, 2, 3	syl2anc 409	. . . 4 ⊢ (𝜑 → (𝑁 ∈ (ℤ≥‘𝑀) ↔ 𝑀 ≤ 𝑁))
5	4	biimpar 295	. . 3 ⊢ ((𝜑 ∧ 𝑀 ≤ 𝑁) → 𝑁 ∈ (ℤ≥‘𝑀))
6		simpl 108	. . 3 ⊢ ((𝜑 ∧ 𝑀 ≤ 𝑁) → 𝜑)
7		eluzfz2 9967	. . . 4 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁))
8		oveq2 5850	. . . . . . . 8 ⊢ (𝑤 = 𝑀 → (𝑀...𝑤) = (𝑀...𝑀))
9	8	rexeqdv 2668	. . . . . . 7 ⊢ (𝑤 = 𝑀 → (∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ ∃𝑛 ∈ (𝑀...𝑀)𝜓))
10	9	dcbid 828	. . . . . 6 ⊢ (𝑤 = 𝑀 → (DECID ∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ DECID ∃𝑛 ∈ (𝑀...𝑀)𝜓))
11	10	imbi2d 229	. . . . 5 ⊢ (𝑤 = 𝑀 → ((𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑤)𝜓) ↔ (𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑀)𝜓)))
12		oveq2 5850	. . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝑀...𝑤) = (𝑀...𝑦))
13	12	rexeqdv 2668	. . . . . . 7 ⊢ (𝑤 = 𝑦 → (∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ ∃𝑛 ∈ (𝑀...𝑦)𝜓))
14	13	dcbid 828	. . . . . 6 ⊢ (𝑤 = 𝑦 → (DECID ∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓))
15	14	imbi2d 229	. . . . 5 ⊢ (𝑤 = 𝑦 → ((𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑤)𝜓) ↔ (𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓)))
16		oveq2 5850	. . . . . . . 8 ⊢ (𝑤 = (𝑦 + 1) → (𝑀...𝑤) = (𝑀...(𝑦 + 1)))
17	16	rexeqdv 2668	. . . . . . 7 ⊢ (𝑤 = (𝑦 + 1) → (∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ ∃𝑛 ∈ (𝑀...(𝑦 + 1))𝜓))
18	17	dcbid 828	. . . . . 6 ⊢ (𝑤 = (𝑦 + 1) → (DECID ∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ DECID ∃𝑛 ∈ (𝑀...(𝑦 + 1))𝜓))
19	18	imbi2d 229	. . . . 5 ⊢ (𝑤 = (𝑦 + 1) → ((𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑤)𝜓) ↔ (𝜑 → DECID ∃𝑛 ∈ (𝑀...(𝑦 + 1))𝜓)))
20		oveq2 5850	. . . . . . . 8 ⊢ (𝑤 = 𝑁 → (𝑀...𝑤) = (𝑀...𝑁))
21	20	rexeqdv 2668	. . . . . . 7 ⊢ (𝑤 = 𝑁 → (∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ ∃𝑛 ∈ (𝑀...𝑁)𝜓))
22	21	dcbid 828	. . . . . 6 ⊢ (𝑤 = 𝑁 → (DECID ∃𝑛 ∈ (𝑀...𝑤)𝜓 ↔ DECID ∃𝑛 ∈ (𝑀...𝑁)𝜓))
23	22	imbi2d 229	. . . . 5 ⊢ (𝑤 = 𝑁 → ((𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑤)𝜓) ↔ (𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑁)𝜓)))
24		eluzfz1 9966	. . . . . . . . 9 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁))
25	24	adantl 275	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈ (𝑀...𝑁))
26		exfzdc.3	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑁)) → DECID 𝜓)
27	26	ralrimiva 2539	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑛 ∈ (𝑀...𝑁)DECID 𝜓)
28	27	adantr 274	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → ∀𝑛 ∈ (𝑀...𝑁)DECID 𝜓)
29		nfsbc1v 2969	. . . . . . . . . 10 ⊢ Ⅎ𝑛[𝑀 / 𝑛]𝜓
30	29	nfdc 1647	. . . . . . . . 9 ⊢ Ⅎ𝑛DECID [𝑀 / 𝑛]𝜓
31		sbceq1a 2960	. . . . . . . . . 10 ⊢ (𝑛 = 𝑀 → (𝜓 ↔ [𝑀 / 𝑛]𝜓))
32	31	dcbid 828	. . . . . . . . 9 ⊢ (𝑛 = 𝑀 → (DECID 𝜓 ↔ DECID [𝑀 / 𝑛]𝜓))
33	30, 32	rspc 2824	. . . . . . . 8 ⊢ (𝑀 ∈ (𝑀...𝑁) → (∀𝑛 ∈ (𝑀...𝑁)DECID 𝜓 → DECID [𝑀 / 𝑛]𝜓))
34	25, 28, 33	sylc 62	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → DECID [𝑀 / 𝑛]𝜓)
35	1	adantr 274	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈ ℤ)
36		fzsn 10001	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℤ → (𝑀...𝑀) = {𝑀})
37	35, 36	syl 14	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (𝑀...𝑀) = {𝑀})
38	37	rexeqdv 2668	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (∃𝑛 ∈ (𝑀...𝑀)𝜓 ↔ ∃𝑛 ∈ {𝑀}𝜓))
39		rexsns 3615	. . . . . . . . 9 ⊢ (∃𝑛 ∈ {𝑀}𝜓 ↔ [𝑀 / 𝑛]𝜓)
40	38, 39	bitrdi 195	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (∃𝑛 ∈ (𝑀...𝑀)𝜓 ↔ [𝑀 / 𝑛]𝜓))
41	40	dcbid 828	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → (DECID ∃𝑛 ∈ (𝑀...𝑀)𝜓 ↔ DECID [𝑀 / 𝑛]𝜓))
42	34, 41	mpbird 166	. . . . . 6 ⊢ ((𝜑 ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → DECID ∃𝑛 ∈ (𝑀...𝑀)𝜓)
43	42	expcom 115	. . . . 5 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑀)𝜓))
44		simpr 109	. . . . . . . . . 10 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓)
45		fzofzp1 10162	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝑀..^𝑁) → (𝑦 + 1) ∈ (𝑀...𝑁))
46	45	ad2antrr 480	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → (𝑦 + 1) ∈ (𝑀...𝑁))
47	27	ad2antlr 481	. . . . . . . . . . . 12 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → ∀𝑛 ∈ (𝑀...𝑁)DECID 𝜓)
48		nfsbc1v 2969	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑛[(𝑦 + 1) / 𝑛]𝜓
49	48	nfdc 1647	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑛DECID [(𝑦 + 1) / 𝑛]𝜓
50		sbceq1a 2960	. . . . . . . . . . . . . 14 ⊢ (𝑛 = (𝑦 + 1) → (𝜓 ↔ [(𝑦 + 1) / 𝑛]𝜓))
51	50	dcbid 828	. . . . . . . . . . . . 13 ⊢ (𝑛 = (𝑦 + 1) → (DECID 𝜓 ↔ DECID [(𝑦 + 1) / 𝑛]𝜓))
52	49, 51	rspc 2824	. . . . . . . . . . . 12 ⊢ ((𝑦 + 1) ∈ (𝑀...𝑁) → (∀𝑛 ∈ (𝑀...𝑁)DECID 𝜓 → DECID [(𝑦 + 1) / 𝑛]𝜓))
53	46, 47, 52	sylc 62	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → DECID [(𝑦 + 1) / 𝑛]𝜓)
54		rexsns 3615	. . . . . . . . . . . 12 ⊢ (∃𝑛 ∈ {(𝑦 + 1)}𝜓 ↔ [(𝑦 + 1) / 𝑛]𝜓)
55	54	dcbii 830	. . . . . . . . . . 11 ⊢ (DECID ∃𝑛 ∈ {(𝑦 + 1)}𝜓 ↔ DECID [(𝑦 + 1) / 𝑛]𝜓)
56	53, 55	sylibr 133	. . . . . . . . . 10 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → DECID ∃𝑛 ∈ {(𝑦 + 1)}𝜓)
57		dcor 925	. . . . . . . . . 10 ⊢ (DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓 → (DECID ∃𝑛 ∈ {(𝑦 + 1)}𝜓 → DECID (∃𝑛 ∈ (𝑀...𝑦)𝜓 ∨ ∃𝑛 ∈ {(𝑦 + 1)}𝜓)))
58	44, 56, 57	sylc 62	. . . . . . . . 9 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → DECID (∃𝑛 ∈ (𝑀...𝑦)𝜓 ∨ ∃𝑛 ∈ {(𝑦 + 1)}𝜓))
59		rexun 3302	. . . . . . . . . 10 ⊢ (∃𝑛 ∈ ((𝑀...𝑦) ∪ {(𝑦 + 1)})𝜓 ↔ (∃𝑛 ∈ (𝑀...𝑦)𝜓 ∨ ∃𝑛 ∈ {(𝑦 + 1)}𝜓))
60	59	dcbii 830	. . . . . . . . 9 ⊢ (DECID ∃𝑛 ∈ ((𝑀...𝑦) ∪ {(𝑦 + 1)})𝜓 ↔ DECID (∃𝑛 ∈ (𝑀...𝑦)𝜓 ∨ ∃𝑛 ∈ {(𝑦 + 1)}𝜓))
61	58, 60	sylibr 133	. . . . . . . 8 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → DECID ∃𝑛 ∈ ((𝑀...𝑦) ∪ {(𝑦 + 1)})𝜓)
62		elfzouz 10086	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝑀..^𝑁) → 𝑦 ∈ (ℤ≥‘𝑀))
63	62	ad2antrr 480	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → 𝑦 ∈ (ℤ≥‘𝑀))
64		fzsuc 10004	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (ℤ≥‘𝑀) → (𝑀...(𝑦 + 1)) = ((𝑀...𝑦) ∪ {(𝑦 + 1)}))
65	63, 64	syl 14	. . . . . . . . . 10 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → (𝑀...(𝑦 + 1)) = ((𝑀...𝑦) ∪ {(𝑦 + 1)}))
66	65	rexeqdv 2668	. . . . . . . . 9 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → (∃𝑛 ∈ (𝑀...(𝑦 + 1))𝜓 ↔ ∃𝑛 ∈ ((𝑀...𝑦) ∪ {(𝑦 + 1)})𝜓))
67	66	dcbid 828	. . . . . . . 8 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → (DECID ∃𝑛 ∈ (𝑀...(𝑦 + 1))𝜓 ↔ DECID ∃𝑛 ∈ ((𝑀...𝑦) ∪ {(𝑦 + 1)})𝜓))
68	61, 67	mpbird 166	. . . . . . 7 ⊢ (((𝑦 ∈ (𝑀..^𝑁) ∧ 𝜑) ∧ DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → DECID ∃𝑛 ∈ (𝑀...(𝑦 + 1))𝜓)
69	68	exp31 362	. . . . . 6 ⊢ (𝑦 ∈ (𝑀..^𝑁) → (𝜑 → (DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓 → DECID ∃𝑛 ∈ (𝑀...(𝑦 + 1))𝜓)))
70	69	a2d 26	. . . . 5 ⊢ (𝑦 ∈ (𝑀..^𝑁) → ((𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑦)𝜓) → (𝜑 → DECID ∃𝑛 ∈ (𝑀...(𝑦 + 1))𝜓)))
71	11, 15, 19, 23, 43, 70	fzind2 10174	. . . 4 ⊢ (𝑁 ∈ (𝑀...𝑁) → (𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑁)𝜓))
72	7, 71	syl 14	. . 3 ⊢ (𝑁 ∈ (ℤ≥‘𝑀) → (𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑁)𝜓))
73	5, 6, 72	sylc 62	. 2 ⊢ ((𝜑 ∧ 𝑀 ≤ 𝑁) → DECID ∃𝑛 ∈ (𝑀...𝑁)𝜓)
74		rex0 3426	. . . . 5 ⊢ ¬ ∃𝑛 ∈ ∅ 𝜓
75		zltnle 9237	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) → (𝑁 < 𝑀 ↔ ¬ 𝑀 ≤ 𝑁))
76	2, 1, 75	syl2anc 409	. . . . . . . 8 ⊢ (𝜑 → (𝑁 < 𝑀 ↔ ¬ 𝑀 ≤ 𝑁))
77	76	biimpar 295	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑀 ≤ 𝑁) → 𝑁 < 𝑀)
78		fzn 9977	. . . . . . . . 9 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅))
79	1, 2, 78	syl2anc 409	. . . . . . . 8 ⊢ (𝜑 → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅))
80	79	adantr 274	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑀 ≤ 𝑁) → (𝑁 < 𝑀 ↔ (𝑀...𝑁) = ∅))
81	77, 80	mpbid 146	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑀 ≤ 𝑁) → (𝑀...𝑁) = ∅)
82	81	rexeqdv 2668	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝑀 ≤ 𝑁) → (∃𝑛 ∈ (𝑀...𝑁)𝜓 ↔ ∃𝑛 ∈ ∅ 𝜓))
83	74, 82	mtbiri 665	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝑀 ≤ 𝑁) → ¬ ∃𝑛 ∈ (𝑀...𝑁)𝜓)
84	83	olcd 724	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑀 ≤ 𝑁) → (∃𝑛 ∈ (𝑀...𝑁)𝜓 ∨ ¬ ∃𝑛 ∈ (𝑀...𝑁)𝜓))
85		df-dc 825	. . 3 ⊢ (DECID ∃𝑛 ∈ (𝑀...𝑁)𝜓 ↔ (∃𝑛 ∈ (𝑀...𝑁)𝜓 ∨ ¬ ∃𝑛 ∈ (𝑀...𝑁)𝜓))
86	84, 85	sylibr 133	. 2 ⊢ ((𝜑 ∧ ¬ 𝑀 ≤ 𝑁) → DECID ∃𝑛 ∈ (𝑀...𝑁)𝜓)
87		zdcle 9267	. . . 4 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → DECID 𝑀 ≤ 𝑁)
88		exmiddc 826	. . . 4 ⊢ (DECID 𝑀 ≤ 𝑁 → (𝑀 ≤ 𝑁 ∨ ¬ 𝑀 ≤ 𝑁))
89	87, 88	syl 14	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ∨ ¬ 𝑀 ≤ 𝑁))
90	1, 2, 89	syl2anc 409	. 2 ⊢ (𝜑 → (𝑀 ≤ 𝑁 ∨ ¬ 𝑀 ≤ 𝑁))
91	73, 86, 90	mpjaodan 788	1 ⊢ (𝜑 → DECID ∃𝑛 ∈ (𝑀...𝑁)𝜓)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698  DECID wdc 824   = wceq 1343   ∈ wcel 2136  ∀wral 2444  ∃wrex 2445  [wsbc 2951   ∪ cun 3114  ∅c0 3409  {csn 3576   class class class wbr 3982  ‘cfv 5188  (class class class)co 5842  1c1 7754   + caddc 7756   < clt 7933   ≤ cle 7934  ℤcz 9191  ℤ_≥cuz 9466  ...cfz 9944  ..^cfzo 10077

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-0id 7861  ax-rnegex 7862  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-inn 8858  df-n0 9115  df-z 9192  df-uz 9467  df-fz 9945  df-fzo 10078

This theorem is referenced by:  prmind2  12052