Theorem infm3 10861

Description: The completeness axiom for reals in terms of infimum: a nonempty, bounded-below set of reals has an infimum. (This theorem is the dual of sup3 10859.) (Contributed by NM, 14-Jun-2005.)

Assertion

Ref	Expression
infm3	⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))

Distinct variable group:   𝑥,𝑦,𝑧,𝐴

Proof of Theorem infm3

Dummy variables 𝑤 𝑣 𝑢 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3562	. . . . . . . . 9 ⊢ (𝐴 ⊆ ℝ → (𝑣 ∈ 𝐴 → 𝑣 ∈ ℝ))
2	1	pm4.71rd 665	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ → (𝑣 ∈ 𝐴 ↔ (𝑣 ∈ ℝ ∧ 𝑣 ∈ 𝐴)))
3	2	exbidv 1837	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ → (∃𝑣 𝑣 ∈ 𝐴 ↔ ∃𝑣(𝑣 ∈ ℝ ∧ 𝑣 ∈ 𝐴)))
4		df-rex 2902	. . . . . . . 8 ⊢ (∃𝑣 ∈ ℝ 𝑣 ∈ 𝐴 ↔ ∃𝑣(𝑣 ∈ ℝ ∧ 𝑣 ∈ 𝐴))
5		renegcl 10223	. . . . . . . . 9 ⊢ (𝑤 ∈ ℝ → -𝑤 ∈ ℝ)
6		infm3lem 10860	. . . . . . . . 9 ⊢ (𝑣 ∈ ℝ → ∃𝑤 ∈ ℝ 𝑣 = -𝑤)
7		eleq1 2676	. . . . . . . . 9 ⊢ (𝑣 = -𝑤 → (𝑣 ∈ 𝐴 ↔ -𝑤 ∈ 𝐴))
8	5, 6, 7	rexxfr 4814	. . . . . . . 8 ⊢ (∃𝑣 ∈ ℝ 𝑣 ∈ 𝐴 ↔ ∃𝑤 ∈ ℝ -𝑤 ∈ 𝐴)
9	4, 8	bitr3i 265	. . . . . . 7 ⊢ (∃𝑣(𝑣 ∈ ℝ ∧ 𝑣 ∈ 𝐴) ↔ ∃𝑤 ∈ ℝ -𝑤 ∈ 𝐴)
10	3, 9	syl6bb 275	. . . . . 6 ⊢ (𝐴 ⊆ ℝ → (∃𝑣 𝑣 ∈ 𝐴 ↔ ∃𝑤 ∈ ℝ -𝑤 ∈ 𝐴))
11		n0 3890	. . . . . 6 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑣 𝑣 ∈ 𝐴)
12		rabn0 3912	. . . . . 6 ⊢ ({𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ≠ ∅ ↔ ∃𝑤 ∈ ℝ -𝑤 ∈ 𝐴)
13	10, 11, 12	3bitr4g 302	. . . . 5 ⊢ (𝐴 ⊆ ℝ → (𝐴 ≠ ∅ ↔ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ≠ ∅))
14		ssel 3562	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ ℝ → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ))
15	14	pm4.71rd 665	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ ℝ → (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴)))
16	15	imbi1d 330	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ℝ → ((𝑦 ∈ 𝐴 → 𝑥 ≤ 𝑦) ↔ ((𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴) → 𝑥 ≤ 𝑦)))
17		impexp 461	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴) → 𝑥 ≤ 𝑦) ↔ (𝑦 ∈ ℝ → (𝑦 ∈ 𝐴 → 𝑥 ≤ 𝑦)))
18	16, 17	syl6bb 275	. . . . . . . . 9 ⊢ (𝐴 ⊆ ℝ → ((𝑦 ∈ 𝐴 → 𝑥 ≤ 𝑦) ↔ (𝑦 ∈ ℝ → (𝑦 ∈ 𝐴 → 𝑥 ≤ 𝑦))))
19	18	albidv 1836	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ → (∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ≤ 𝑦) ↔ ∀𝑦(𝑦 ∈ ℝ → (𝑦 ∈ 𝐴 → 𝑥 ≤ 𝑦))))
20		df-ral 2901	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ≤ 𝑦))
21		renegcl 10223	. . . . . . . . . 10 ⊢ (𝑣 ∈ ℝ → -𝑣 ∈ ℝ)
22		infm3lem 10860	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → ∃𝑣 ∈ ℝ 𝑦 = -𝑣)
23		eleq1 2676	. . . . . . . . . . 11 ⊢ (𝑦 = -𝑣 → (𝑦 ∈ 𝐴 ↔ -𝑣 ∈ 𝐴))
24		breq2 4587	. . . . . . . . . . 11 ⊢ (𝑦 = -𝑣 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ -𝑣))
25	23, 24	imbi12d 333	. . . . . . . . . 10 ⊢ (𝑦 = -𝑣 → ((𝑦 ∈ 𝐴 → 𝑥 ≤ 𝑦) ↔ (-𝑣 ∈ 𝐴 → 𝑥 ≤ -𝑣)))
26	21, 22, 25	ralxfr 4812	. . . . . . . . 9 ⊢ (∀𝑦 ∈ ℝ (𝑦 ∈ 𝐴 → 𝑥 ≤ 𝑦) ↔ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → 𝑥 ≤ -𝑣))
27		df-ral 2901	. . . . . . . . 9 ⊢ (∀𝑦 ∈ ℝ (𝑦 ∈ 𝐴 → 𝑥 ≤ 𝑦) ↔ ∀𝑦(𝑦 ∈ ℝ → (𝑦 ∈ 𝐴 → 𝑥 ≤ 𝑦)))
28	26, 27	bitr3i 265	. . . . . . . 8 ⊢ (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → 𝑥 ≤ -𝑣) ↔ ∀𝑦(𝑦 ∈ ℝ → (𝑦 ∈ 𝐴 → 𝑥 ≤ 𝑦)))
29	19, 20, 28	3bitr4g 302	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ → (∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → 𝑥 ≤ -𝑣)))
30	29	rexbidv 3034	. . . . . 6 ⊢ (𝐴 ⊆ ℝ → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → 𝑥 ≤ -𝑣)))
31		renegcl 10223	. . . . . . . 8 ⊢ (𝑢 ∈ ℝ → -𝑢 ∈ ℝ)
32		infm3lem 10860	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → ∃𝑢 ∈ ℝ 𝑥 = -𝑢)
33		breq1 4586	. . . . . . . . . 10 ⊢ (𝑥 = -𝑢 → (𝑥 ≤ -𝑣 ↔ -𝑢 ≤ -𝑣))
34	33	imbi2d 329	. . . . . . . . 9 ⊢ (𝑥 = -𝑢 → ((-𝑣 ∈ 𝐴 → 𝑥 ≤ -𝑣) ↔ (-𝑣 ∈ 𝐴 → -𝑢 ≤ -𝑣)))
35	34	ralbidv 2969	. . . . . . . 8 ⊢ (𝑥 = -𝑢 → (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → 𝑥 ≤ -𝑣) ↔ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → -𝑢 ≤ -𝑣)))
36	31, 32, 35	rexxfr 4814	. . . . . . 7 ⊢ (∃𝑥 ∈ ℝ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → 𝑥 ≤ -𝑣) ↔ ∃𝑢 ∈ ℝ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → -𝑢 ≤ -𝑣))
37		negeq 10152	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑣 → -𝑤 = -𝑣)
38	37	eleq1d 2672	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑣 → (-𝑤 ∈ 𝐴 ↔ -𝑣 ∈ 𝐴))
39	38	elrab 3331	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ↔ (𝑣 ∈ ℝ ∧ -𝑣 ∈ 𝐴))
40	39	imbi1i 338	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} → 𝑣 ≤ 𝑢) ↔ ((𝑣 ∈ ℝ ∧ -𝑣 ∈ 𝐴) → 𝑣 ≤ 𝑢))
41		impexp 461	. . . . . . . . . . . 12 ⊢ (((𝑣 ∈ ℝ ∧ -𝑣 ∈ 𝐴) → 𝑣 ≤ 𝑢) ↔ (𝑣 ∈ ℝ → (-𝑣 ∈ 𝐴 → 𝑣 ≤ 𝑢)))
42	40, 41	bitri 263	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} → 𝑣 ≤ 𝑢) ↔ (𝑣 ∈ ℝ → (-𝑣 ∈ 𝐴 → 𝑣 ≤ 𝑢)))
43	42	albii 1737	. . . . . . . . . 10 ⊢ (∀𝑣(𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} → 𝑣 ≤ 𝑢) ↔ ∀𝑣(𝑣 ∈ ℝ → (-𝑣 ∈ 𝐴 → 𝑣 ≤ 𝑢)))
44		df-ral 2901	. . . . . . . . . 10 ⊢ (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 ≤ 𝑢 ↔ ∀𝑣(𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} → 𝑣 ≤ 𝑢))
45		df-ral 2901	. . . . . . . . . 10 ⊢ (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → 𝑣 ≤ 𝑢) ↔ ∀𝑣(𝑣 ∈ ℝ → (-𝑣 ∈ 𝐴 → 𝑣 ≤ 𝑢)))
46	43, 44, 45	3bitr4ri 292	. . . . . . . . 9 ⊢ (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → 𝑣 ≤ 𝑢) ↔ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 ≤ 𝑢)
47		leneg 10410	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑣 ≤ 𝑢 ↔ -𝑢 ≤ -𝑣))
48	47	ancoms 468	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑣 ≤ 𝑢 ↔ -𝑢 ≤ -𝑣))
49	48	imbi2d 329	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((-𝑣 ∈ 𝐴 → 𝑣 ≤ 𝑢) ↔ (-𝑣 ∈ 𝐴 → -𝑢 ≤ -𝑣)))
50	49	ralbidva 2968	. . . . . . . . 9 ⊢ (𝑢 ∈ ℝ → (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → 𝑣 ≤ 𝑢) ↔ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → -𝑢 ≤ -𝑣)))
51	46, 50	syl5bbr 273	. . . . . . . 8 ⊢ (𝑢 ∈ ℝ → (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 ≤ 𝑢 ↔ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → -𝑢 ≤ -𝑣)))
52	51	rexbiia 3022	. . . . . . 7 ⊢ (∃𝑢 ∈ ℝ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 ≤ 𝑢 ↔ ∃𝑢 ∈ ℝ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → -𝑢 ≤ -𝑣))
53	36, 52	bitr4i 266	. . . . . 6 ⊢ (∃𝑥 ∈ ℝ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → 𝑥 ≤ -𝑣) ↔ ∃𝑢 ∈ ℝ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 ≤ 𝑢)
54	30, 53	syl6bb 275	. . . . 5 ⊢ (𝐴 ⊆ ℝ → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ ∃𝑢 ∈ ℝ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 ≤ 𝑢))
55	13, 54	anbi12d 743	. . . 4 ⊢ (𝐴 ⊆ ℝ → ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) ↔ ({𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ≠ ∅ ∧ ∃𝑢 ∈ ℝ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 ≤ 𝑢)))
56		ssrab2 3650	. . . . 5 ⊢ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ⊆ ℝ
57		sup3 10859	. . . . 5 ⊢ (({𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ⊆ ℝ ∧ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ≠ ∅ ∧ ∃𝑢 ∈ ℝ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 ≤ 𝑢) → ∃𝑢 ∈ ℝ (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑢 < 𝑣 ∧ ∀𝑣 ∈ ℝ (𝑣 < 𝑢 → ∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 < 𝑡)))
58	56, 57	mp3an1 1403	. . . 4 ⊢ (({𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ≠ ∅ ∧ ∃𝑢 ∈ ℝ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 ≤ 𝑢) → ∃𝑢 ∈ ℝ (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑢 < 𝑣 ∧ ∀𝑣 ∈ ℝ (𝑣 < 𝑢 → ∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 < 𝑡)))
59	55, 58	syl6bi 242	. . 3 ⊢ (𝐴 ⊆ ℝ → ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∃𝑢 ∈ ℝ (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑢 < 𝑣 ∧ ∀𝑣 ∈ ℝ (𝑣 < 𝑢 → ∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 < 𝑡))))
60	15	imbi1d 330	. . . . . . . . 9 ⊢ (𝐴 ⊆ ℝ → ((𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥) ↔ ((𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴) → ¬ 𝑦 < 𝑥)))
61		impexp 461	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴) → ¬ 𝑦 < 𝑥) ↔ (𝑦 ∈ ℝ → (𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥)))
62	60, 61	syl6bb 275	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ → ((𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥) ↔ (𝑦 ∈ ℝ → (𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥))))
63	62	albidv 1836	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ → (∀𝑦(𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥) ↔ ∀𝑦(𝑦 ∈ ℝ → (𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥))))
64		df-ral 2901	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥))
65		breq1 4586	. . . . . . . . . . 11 ⊢ (𝑦 = -𝑣 → (𝑦 < 𝑥 ↔ -𝑣 < 𝑥))
66	65	notbid 307	. . . . . . . . . 10 ⊢ (𝑦 = -𝑣 → (¬ 𝑦 < 𝑥 ↔ ¬ -𝑣 < 𝑥))
67	23, 66	imbi12d 333	. . . . . . . . 9 ⊢ (𝑦 = -𝑣 → ((𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥) ↔ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < 𝑥)))
68	21, 22, 67	ralxfr 4812	. . . . . . . 8 ⊢ (∀𝑦 ∈ ℝ (𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥) ↔ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < 𝑥))
69		df-ral 2901	. . . . . . . 8 ⊢ (∀𝑦 ∈ ℝ (𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥) ↔ ∀𝑦(𝑦 ∈ ℝ → (𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥)))
70	68, 69	bitr3i 265	. . . . . . 7 ⊢ (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < 𝑥) ↔ ∀𝑦(𝑦 ∈ ℝ → (𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥)))
71	63, 64, 70	3bitr4g 302	. . . . . 6 ⊢ (𝐴 ⊆ ℝ → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < 𝑥)))
72		breq2 4587	. . . . . . . . 9 ⊢ (𝑦 = -𝑣 → (𝑥 < 𝑦 ↔ 𝑥 < -𝑣))
73		breq2 4587	. . . . . . . . . 10 ⊢ (𝑦 = -𝑣 → (𝑧 < 𝑦 ↔ 𝑧 < -𝑣))
74	73	rexbidv 3034	. . . . . . . . 9 ⊢ (𝑦 = -𝑣 → (∃𝑧 ∈ 𝐴 𝑧 < 𝑦 ↔ ∃𝑧 ∈ 𝐴 𝑧 < -𝑣))
75	72, 74	imbi12d 333	. . . . . . . 8 ⊢ (𝑦 = -𝑣 → ((𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ (𝑥 < -𝑣 → ∃𝑧 ∈ 𝐴 𝑧 < -𝑣)))
76	21, 22, 75	ralxfr 4812	. . . . . . 7 ⊢ (∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ ∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑧 ∈ 𝐴 𝑧 < -𝑣))
77		ssel 3562	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ ℝ → (𝑧 ∈ 𝐴 → 𝑧 ∈ ℝ))
78	77	adantrd 483	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ ℝ → ((𝑧 ∈ 𝐴 ∧ 𝑧 < -𝑣) → 𝑧 ∈ ℝ))
79	78	pm4.71rd 665	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ ℝ → ((𝑧 ∈ 𝐴 ∧ 𝑧 < -𝑣) ↔ (𝑧 ∈ ℝ ∧ (𝑧 ∈ 𝐴 ∧ 𝑧 < -𝑣))))
80	79	exbidv 1837	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ℝ → (∃𝑧(𝑧 ∈ 𝐴 ∧ 𝑧 < -𝑣) ↔ ∃𝑧(𝑧 ∈ ℝ ∧ (𝑧 ∈ 𝐴 ∧ 𝑧 < -𝑣))))
81		df-rex 2902	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝐴 𝑧 < -𝑣 ↔ ∃𝑧(𝑧 ∈ 𝐴 ∧ 𝑧 < -𝑣))
82		renegcl 10223	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ℝ → -𝑡 ∈ ℝ)
83		infm3lem 10860	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℝ → ∃𝑡 ∈ ℝ 𝑧 = -𝑡)
84		eleq1 2676	. . . . . . . . . . . . 13 ⊢ (𝑧 = -𝑡 → (𝑧 ∈ 𝐴 ↔ -𝑡 ∈ 𝐴))
85		breq1 4586	. . . . . . . . . . . . 13 ⊢ (𝑧 = -𝑡 → (𝑧 < -𝑣 ↔ -𝑡 < -𝑣))
86	84, 85	anbi12d 743	. . . . . . . . . . . 12 ⊢ (𝑧 = -𝑡 → ((𝑧 ∈ 𝐴 ∧ 𝑧 < -𝑣) ↔ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣)))
87	82, 83, 86	rexxfr 4814	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ ℝ (𝑧 ∈ 𝐴 ∧ 𝑧 < -𝑣) ↔ ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣))
88		df-rex 2902	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ ℝ (𝑧 ∈ 𝐴 ∧ 𝑧 < -𝑣) ↔ ∃𝑧(𝑧 ∈ ℝ ∧ (𝑧 ∈ 𝐴 ∧ 𝑧 < -𝑣)))
89	87, 88	bitr3i 265	. . . . . . . . . 10 ⊢ (∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣) ↔ ∃𝑧(𝑧 ∈ ℝ ∧ (𝑧 ∈ 𝐴 ∧ 𝑧 < -𝑣)))
90	80, 81, 89	3bitr4g 302	. . . . . . . . 9 ⊢ (𝐴 ⊆ ℝ → (∃𝑧 ∈ 𝐴 𝑧 < -𝑣 ↔ ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣)))
91	90	imbi2d 329	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ → ((𝑥 < -𝑣 → ∃𝑧 ∈ 𝐴 𝑧 < -𝑣) ↔ (𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣))))
92	91	ralbidv 2969	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ → (∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑧 ∈ 𝐴 𝑧 < -𝑣) ↔ ∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣))))
93	76, 92	syl5bb 271	. . . . . 6 ⊢ (𝐴 ⊆ ℝ → (∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ ∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣))))
94	71, 93	anbi12d 743	. . . . 5 ⊢ (𝐴 ⊆ ℝ → ((∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) ↔ (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < 𝑥) ∧ ∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣)))))
95	94	rexbidv 3034	. . . 4 ⊢ (𝐴 ⊆ ℝ → (∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) ↔ ∃𝑥 ∈ ℝ (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < 𝑥) ∧ ∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣)))))
96		breq2 4587	. . . . . . . . . 10 ⊢ (𝑥 = -𝑢 → (-𝑣 < 𝑥 ↔ -𝑣 < -𝑢))
97	96	notbid 307	. . . . . . . . 9 ⊢ (𝑥 = -𝑢 → (¬ -𝑣 < 𝑥 ↔ ¬ -𝑣 < -𝑢))
98	97	imbi2d 329	. . . . . . . 8 ⊢ (𝑥 = -𝑢 → ((-𝑣 ∈ 𝐴 → ¬ -𝑣 < 𝑥) ↔ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < -𝑢)))
99	98	ralbidv 2969	. . . . . . 7 ⊢ (𝑥 = -𝑢 → (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < 𝑥) ↔ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < -𝑢)))
100		breq1 4586	. . . . . . . . 9 ⊢ (𝑥 = -𝑢 → (𝑥 < -𝑣 ↔ -𝑢 < -𝑣))
101	100	imbi1d 330	. . . . . . . 8 ⊢ (𝑥 = -𝑢 → ((𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣)) ↔ (-𝑢 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣))))
102	101	ralbidv 2969	. . . . . . 7 ⊢ (𝑥 = -𝑢 → (∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣)) ↔ ∀𝑣 ∈ ℝ (-𝑢 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣))))
103	99, 102	anbi12d 743	. . . . . 6 ⊢ (𝑥 = -𝑢 → ((∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < 𝑥) ∧ ∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣))) ↔ (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < -𝑢) ∧ ∀𝑣 ∈ ℝ (-𝑢 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣)))))
104	31, 32, 103	rexxfr 4814	. . . . 5 ⊢ (∃𝑥 ∈ ℝ (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < 𝑥) ∧ ∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣))) ↔ ∃𝑢 ∈ ℝ (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < -𝑢) ∧ ∀𝑣 ∈ ℝ (-𝑢 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣))))
105	39	imbi1i 338	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} → ¬ 𝑢 < 𝑣) ↔ ((𝑣 ∈ ℝ ∧ -𝑣 ∈ 𝐴) → ¬ 𝑢 < 𝑣))
106		impexp 461	. . . . . . . . . . 11 ⊢ (((𝑣 ∈ ℝ ∧ -𝑣 ∈ 𝐴) → ¬ 𝑢 < 𝑣) ↔ (𝑣 ∈ ℝ → (-𝑣 ∈ 𝐴 → ¬ 𝑢 < 𝑣)))
107	105, 106	bitri 263	. . . . . . . . . 10 ⊢ ((𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} → ¬ 𝑢 < 𝑣) ↔ (𝑣 ∈ ℝ → (-𝑣 ∈ 𝐴 → ¬ 𝑢 < 𝑣)))
108	107	albii 1737	. . . . . . . . 9 ⊢ (∀𝑣(𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} → ¬ 𝑢 < 𝑣) ↔ ∀𝑣(𝑣 ∈ ℝ → (-𝑣 ∈ 𝐴 → ¬ 𝑢 < 𝑣)))
109		df-ral 2901	. . . . . . . . 9 ⊢ (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑢 < 𝑣 ↔ ∀𝑣(𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} → ¬ 𝑢 < 𝑣))
110		df-ral 2901	. . . . . . . . 9 ⊢ (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ 𝑢 < 𝑣) ↔ ∀𝑣(𝑣 ∈ ℝ → (-𝑣 ∈ 𝐴 → ¬ 𝑢 < 𝑣)))
111	108, 109, 110	3bitr4ri 292	. . . . . . . 8 ⊢ (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ 𝑢 < 𝑣) ↔ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑢 < 𝑣)
112		ltneg 10407	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢 < 𝑣 ↔ -𝑣 < -𝑢))
113	112	notbid 307	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (¬ 𝑢 < 𝑣 ↔ ¬ -𝑣 < -𝑢))
114	113	imbi2d 329	. . . . . . . . 9 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((-𝑣 ∈ 𝐴 → ¬ 𝑢 < 𝑣) ↔ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < -𝑢)))
115	114	ralbidva 2968	. . . . . . . 8 ⊢ (𝑢 ∈ ℝ → (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ 𝑢 < 𝑣) ↔ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < -𝑢)))
116	111, 115	syl5bbr 273	. . . . . . 7 ⊢ (𝑢 ∈ ℝ → (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑢 < 𝑣 ↔ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < -𝑢)))
117		ltneg 10407	. . . . . . . . . 10 ⊢ ((𝑣 ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑣 < 𝑢 ↔ -𝑢 < -𝑣))
118	117	ancoms 468	. . . . . . . . 9 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑣 < 𝑢 ↔ -𝑢 < -𝑣))
119		negeq 10152	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑡 → -𝑤 = -𝑡)
120	119	eleq1d 2672	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑡 → (-𝑤 ∈ 𝐴 ↔ -𝑡 ∈ 𝐴))
121	120	rexrab 3337	. . . . . . . . . . 11 ⊢ (∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 < 𝑡 ↔ ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ 𝑣 < 𝑡))
122		ltneg 10407	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑣 < 𝑡 ↔ -𝑡 < -𝑣))
123	122	anbi2d 736	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ ℝ ∧ 𝑡 ∈ ℝ) → ((-𝑡 ∈ 𝐴 ∧ 𝑣 < 𝑡) ↔ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣)))
124	123	rexbidva 3031	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ℝ → (∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ 𝑣 < 𝑡) ↔ ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣)))
125	121, 124	syl5bb 271	. . . . . . . . . 10 ⊢ (𝑣 ∈ ℝ → (∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 < 𝑡 ↔ ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣)))
126	125	adantl 481	. . . . . . . . 9 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 < 𝑡 ↔ ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣)))
127	118, 126	imbi12d 333	. . . . . . . 8 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((𝑣 < 𝑢 → ∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 < 𝑡) ↔ (-𝑢 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣))))
128	127	ralbidva 2968	. . . . . . 7 ⊢ (𝑢 ∈ ℝ → (∀𝑣 ∈ ℝ (𝑣 < 𝑢 → ∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 < 𝑡) ↔ ∀𝑣 ∈ ℝ (-𝑢 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣))))
129	116, 128	anbi12d 743	. . . . . 6 ⊢ (𝑢 ∈ ℝ → ((∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑢 < 𝑣 ∧ ∀𝑣 ∈ ℝ (𝑣 < 𝑢 → ∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 < 𝑡)) ↔ (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < -𝑢) ∧ ∀𝑣 ∈ ℝ (-𝑢 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣)))))
130	129	rexbiia 3022	. . . . 5 ⊢ (∃𝑢 ∈ ℝ (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑢 < 𝑣 ∧ ∀𝑣 ∈ ℝ (𝑣 < 𝑢 → ∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 < 𝑡)) ↔ ∃𝑢 ∈ ℝ (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < -𝑢) ∧ ∀𝑣 ∈ ℝ (-𝑢 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣))))
131	104, 130	bitr4i 266	. . . 4 ⊢ (∃𝑥 ∈ ℝ (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < 𝑥) ∧ ∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣))) ↔ ∃𝑢 ∈ ℝ (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑢 < 𝑣 ∧ ∀𝑣 ∈ ℝ (𝑣 < 𝑢 → ∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 < 𝑡)))
132	95, 131	syl6bb 275	. . 3 ⊢ (𝐴 ⊆ ℝ → (∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)) ↔ ∃𝑢 ∈ ℝ (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑢 < 𝑣 ∧ ∀𝑣 ∈ ℝ (𝑣 < 𝑢 → ∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 < 𝑡))))
133	59, 132	sylibrd 248	. 2 ⊢ (𝐴 ⊆ ℝ → ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦))))
134	133	3impib 1254	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900   ⊆ wss 3540  ∅c0 3874   class class class wbr 4583  ℝcr 9814   < clt 9953   ≤ cle 9954  -cneg 10146

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-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892  ax-pre-sup 9893

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  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-nel 2783  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-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  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-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148

This theorem is referenced by:  infrecl  10882  infrenegsup  10883  infregelb  10884  infrelb  10885  xrinfmsslem  12010  gtinf  31483  gtinfOLD  31484  infrglb  38657