Theorem infm3 12180

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 12178.) (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 3975	. . . . . . . . 9 ⊢ (𝐴 ⊆ ℝ → (𝑣 ∈ 𝐴 → 𝑣 ∈ ℝ))
2	1	pm4.71rd 562	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ → (𝑣 ∈ 𝐴 ↔ (𝑣 ∈ ℝ ∧ 𝑣 ∈ 𝐴)))
3	2	exbidv 1923	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ → (∃𝑣 𝑣 ∈ 𝐴 ↔ ∃𝑣(𝑣 ∈ ℝ ∧ 𝑣 ∈ 𝐴)))
4		df-rex 3070	. . . . . . . 8 ⊢ (∃𝑣 ∈ ℝ 𝑣 ∈ 𝐴 ↔ ∃𝑣(𝑣 ∈ ℝ ∧ 𝑣 ∈ 𝐴))
5		renegcl 11530	. . . . . . . . 9 ⊢ (𝑤 ∈ ℝ → -𝑤 ∈ ℝ)
6		infm3lem 12179	. . . . . . . . 9 ⊢ (𝑣 ∈ ℝ → ∃𝑤 ∈ ℝ 𝑣 = -𝑤)
7		eleq1 2820	. . . . . . . . 9 ⊢ (𝑣 = -𝑤 → (𝑣 ∈ 𝐴 ↔ -𝑤 ∈ 𝐴))
8	5, 6, 7	rexxfr 5414	. . . . . . . 8 ⊢ (∃𝑣 ∈ ℝ 𝑣 ∈ 𝐴 ↔ ∃𝑤 ∈ ℝ -𝑤 ∈ 𝐴)
9	4, 8	bitr3i 277	. . . . . . 7 ⊢ (∃𝑣(𝑣 ∈ ℝ ∧ 𝑣 ∈ 𝐴) ↔ ∃𝑤 ∈ ℝ -𝑤 ∈ 𝐴)
10	3, 9	bitrdi 287	. . . . . 6 ⊢ (𝐴 ⊆ ℝ → (∃𝑣 𝑣 ∈ 𝐴 ↔ ∃𝑤 ∈ ℝ -𝑤 ∈ 𝐴))
11		n0 4346	. . . . . 6 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑣 𝑣 ∈ 𝐴)
12		rabn0 4385	. . . . . 6 ⊢ ({𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ≠ ∅ ↔ ∃𝑤 ∈ ℝ -𝑤 ∈ 𝐴)
13	10, 11, 12	3bitr4g 314	. . . . 5 ⊢ (𝐴 ⊆ ℝ → (𝐴 ≠ ∅ ↔ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ≠ ∅))
14		ssel 3975	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ ℝ → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ))
15	14	pm4.71rd 562	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ ℝ → (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴)))
16	15	imbi1d 341	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ℝ → ((𝑦 ∈ 𝐴 → 𝑥 ≤ 𝑦) ↔ ((𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴) → 𝑥 ≤ 𝑦)))
17		impexp 450	. . . . . . . . . 10 ⊢ (((𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴) → 𝑥 ≤ 𝑦) ↔ (𝑦 ∈ ℝ → (𝑦 ∈ 𝐴 → 𝑥 ≤ 𝑦)))
18	16, 17	bitrdi 287	. . . . . . . . 9 ⊢ (𝐴 ⊆ ℝ → ((𝑦 ∈ 𝐴 → 𝑥 ≤ 𝑦) ↔ (𝑦 ∈ ℝ → (𝑦 ∈ 𝐴 → 𝑥 ≤ 𝑦))))
19	18	albidv 1922	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ → (∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ≤ 𝑦) ↔ ∀𝑦(𝑦 ∈ ℝ → (𝑦 ∈ 𝐴 → 𝑥 ≤ 𝑦))))
20		df-ral 3061	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ ∀𝑦(𝑦 ∈ 𝐴 → 𝑥 ≤ 𝑦))
21		renegcl 11530	. . . . . . . . . 10 ⊢ (𝑣 ∈ ℝ → -𝑣 ∈ ℝ)
22		infm3lem 12179	. . . . . . . . . 10 ⊢ (𝑦 ∈ ℝ → ∃𝑣 ∈ ℝ 𝑦 = -𝑣)
23		eleq1 2820	. . . . . . . . . . 11 ⊢ (𝑦 = -𝑣 → (𝑦 ∈ 𝐴 ↔ -𝑣 ∈ 𝐴))
24		breq2 5152	. . . . . . . . . . 11 ⊢ (𝑦 = -𝑣 → (𝑥 ≤ 𝑦 ↔ 𝑥 ≤ -𝑣))
25	23, 24	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑦 = -𝑣 → ((𝑦 ∈ 𝐴 → 𝑥 ≤ 𝑦) ↔ (-𝑣 ∈ 𝐴 → 𝑥 ≤ -𝑣)))
26	21, 22, 25	ralxfr 5412	. . . . . . . . 9 ⊢ (∀𝑦 ∈ ℝ (𝑦 ∈ 𝐴 → 𝑥 ≤ 𝑦) ↔ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → 𝑥 ≤ -𝑣))
27		df-ral 3061	. . . . . . . . 9 ⊢ (∀𝑦 ∈ ℝ (𝑦 ∈ 𝐴 → 𝑥 ≤ 𝑦) ↔ ∀𝑦(𝑦 ∈ ℝ → (𝑦 ∈ 𝐴 → 𝑥 ≤ 𝑦)))
28	26, 27	bitr3i 277	. . . . . . . 8 ⊢ (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → 𝑥 ≤ -𝑣) ↔ ∀𝑦(𝑦 ∈ ℝ → (𝑦 ∈ 𝐴 → 𝑥 ≤ 𝑦)))
29	19, 20, 28	3bitr4g 314	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ → (∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → 𝑥 ≤ -𝑣)))
30	29	rexbidv 3177	. . . . . 6 ⊢ (𝐴 ⊆ ℝ → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → 𝑥 ≤ -𝑣)))
31		renegcl 11530	. . . . . . . 8 ⊢ (𝑢 ∈ ℝ → -𝑢 ∈ ℝ)
32		infm3lem 12179	. . . . . . . 8 ⊢ (𝑥 ∈ ℝ → ∃𝑢 ∈ ℝ 𝑥 = -𝑢)
33		breq1 5151	. . . . . . . . . 10 ⊢ (𝑥 = -𝑢 → (𝑥 ≤ -𝑣 ↔ -𝑢 ≤ -𝑣))
34	33	imbi2d 340	. . . . . . . . 9 ⊢ (𝑥 = -𝑢 → ((-𝑣 ∈ 𝐴 → 𝑥 ≤ -𝑣) ↔ (-𝑣 ∈ 𝐴 → -𝑢 ≤ -𝑣)))
35	34	ralbidv 3176	. . . . . . . 8 ⊢ (𝑥 = -𝑢 → (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → 𝑥 ≤ -𝑣) ↔ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → -𝑢 ≤ -𝑣)))
36	31, 32, 35	rexxfr 5414	. . . . . . 7 ⊢ (∃𝑥 ∈ ℝ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → 𝑥 ≤ -𝑣) ↔ ∃𝑢 ∈ ℝ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → -𝑢 ≤ -𝑣))
37		negeq 11459	. . . . . . . . . . . . . . 15 ⊢ (𝑤 = 𝑣 → -𝑤 = -𝑣)
38	37	eleq1d 2817	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝑣 → (-𝑤 ∈ 𝐴 ↔ -𝑣 ∈ 𝐴))
39	38	elrab 3683	. . . . . . . . . . . . 13 ⊢ (𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ↔ (𝑣 ∈ ℝ ∧ -𝑣 ∈ 𝐴))
40	39	imbi1i 349	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} → 𝑣 ≤ 𝑢) ↔ ((𝑣 ∈ ℝ ∧ -𝑣 ∈ 𝐴) → 𝑣 ≤ 𝑢))
41		impexp 450	. . . . . . . . . . . 12 ⊢ (((𝑣 ∈ ℝ ∧ -𝑣 ∈ 𝐴) → 𝑣 ≤ 𝑢) ↔ (𝑣 ∈ ℝ → (-𝑣 ∈ 𝐴 → 𝑣 ≤ 𝑢)))
42	40, 41	bitri 275	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} → 𝑣 ≤ 𝑢) ↔ (𝑣 ∈ ℝ → (-𝑣 ∈ 𝐴 → 𝑣 ≤ 𝑢)))
43	42	albii 1820	. . . . . . . . . 10 ⊢ (∀𝑣(𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} → 𝑣 ≤ 𝑢) ↔ ∀𝑣(𝑣 ∈ ℝ → (-𝑣 ∈ 𝐴 → 𝑣 ≤ 𝑢)))
44		df-ral 3061	. . . . . . . . . 10 ⊢ (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 ≤ 𝑢 ↔ ∀𝑣(𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} → 𝑣 ≤ 𝑢))
45		df-ral 3061	. . . . . . . . . 10 ⊢ (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → 𝑣 ≤ 𝑢) ↔ ∀𝑣(𝑣 ∈ ℝ → (-𝑣 ∈ 𝐴 → 𝑣 ≤ 𝑢)))
46	43, 44, 45	3bitr4ri 304	. . . . . . . . 9 ⊢ (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → 𝑣 ≤ 𝑢) ↔ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 ≤ 𝑢)
47		leneg 11724	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑣 ≤ 𝑢 ↔ -𝑢 ≤ -𝑣))
48	47	ancoms 458	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑣 ≤ 𝑢 ↔ -𝑢 ≤ -𝑣))
49	48	imbi2d 340	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((-𝑣 ∈ 𝐴 → 𝑣 ≤ 𝑢) ↔ (-𝑣 ∈ 𝐴 → -𝑢 ≤ -𝑣)))
50	49	ralbidva 3174	. . . . . . . . 9 ⊢ (𝑢 ∈ ℝ → (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → 𝑣 ≤ 𝑢) ↔ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → -𝑢 ≤ -𝑣)))
51	46, 50	bitr3id 285	. . . . . . . 8 ⊢ (𝑢 ∈ ℝ → (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 ≤ 𝑢 ↔ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → -𝑢 ≤ -𝑣)))
52	51	rexbiia 3091	. . . . . . 7 ⊢ (∃𝑢 ∈ ℝ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 ≤ 𝑢 ↔ ∃𝑢 ∈ ℝ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → -𝑢 ≤ -𝑣))
53	36, 52	bitr4i 278	. . . . . 6 ⊢ (∃𝑥 ∈ ℝ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → 𝑥 ≤ -𝑣) ↔ ∃𝑢 ∈ ℝ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 ≤ 𝑢)
54	30, 53	bitrdi 287	. . . . 5 ⊢ (𝐴 ⊆ ℝ → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑥 ≤ 𝑦 ↔ ∃𝑢 ∈ ℝ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 ≤ 𝑢))
55	13, 54	anbi12d 630	. . . 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 4077	. . . . 5 ⊢ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ⊆ ℝ
57		sup3 12178	. . . . 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 1447	. . . 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 253	. . 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 341	. . . . . . . . 9 ⊢ (𝐴 ⊆ ℝ → ((𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥) ↔ ((𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴) → ¬ 𝑦 < 𝑥)))
61		impexp 450	. . . . . . . . 9 ⊢ (((𝑦 ∈ ℝ ∧ 𝑦 ∈ 𝐴) → ¬ 𝑦 < 𝑥) ↔ (𝑦 ∈ ℝ → (𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥)))
62	60, 61	bitrdi 287	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ → ((𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥) ↔ (𝑦 ∈ ℝ → (𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥))))
63	62	albidv 1922	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ → (∀𝑦(𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥) ↔ ∀𝑦(𝑦 ∈ ℝ → (𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥))))
64		df-ral 3061	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥))
65		breq1 5151	. . . . . . . . . . 11 ⊢ (𝑦 = -𝑣 → (𝑦 < 𝑥 ↔ -𝑣 < 𝑥))
66	65	notbid 318	. . . . . . . . . 10 ⊢ (𝑦 = -𝑣 → (¬ 𝑦 < 𝑥 ↔ ¬ -𝑣 < 𝑥))
67	23, 66	imbi12d 344	. . . . . . . . 9 ⊢ (𝑦 = -𝑣 → ((𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥) ↔ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < 𝑥)))
68	21, 22, 67	ralxfr 5412	. . . . . . . 8 ⊢ (∀𝑦 ∈ ℝ (𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥) ↔ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < 𝑥))
69		df-ral 3061	. . . . . . . 8 ⊢ (∀𝑦 ∈ ℝ (𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥) ↔ ∀𝑦(𝑦 ∈ ℝ → (𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥)))
70	68, 69	bitr3i 277	. . . . . . 7 ⊢ (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < 𝑥) ↔ ∀𝑦(𝑦 ∈ ℝ → (𝑦 ∈ 𝐴 → ¬ 𝑦 < 𝑥)))
71	63, 64, 70	3bitr4g 314	. . . . . 6 ⊢ (𝐴 ⊆ ℝ → (∀𝑦 ∈ 𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < 𝑥)))
72		breq2 5152	. . . . . . . . 9 ⊢ (𝑦 = -𝑣 → (𝑥 < 𝑦 ↔ 𝑥 < -𝑣))
73		breq2 5152	. . . . . . . . . 10 ⊢ (𝑦 = -𝑣 → (𝑧 < 𝑦 ↔ 𝑧 < -𝑣))
74	73	rexbidv 3177	. . . . . . . . 9 ⊢ (𝑦 = -𝑣 → (∃𝑧 ∈ 𝐴 𝑧 < 𝑦 ↔ ∃𝑧 ∈ 𝐴 𝑧 < -𝑣))
75	72, 74	imbi12d 344	. . . . . . . 8 ⊢ (𝑦 = -𝑣 → ((𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ (𝑥 < -𝑣 → ∃𝑧 ∈ 𝐴 𝑧 < -𝑣)))
76	21, 22, 75	ralxfr 5412	. . . . . . 7 ⊢ (∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ 𝐴 𝑧 < 𝑦) ↔ ∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑧 ∈ 𝐴 𝑧 < -𝑣))
77		ssel 3975	. . . . . . . . . . . . 13 ⊢ (𝐴 ⊆ ℝ → (𝑧 ∈ 𝐴 → 𝑧 ∈ ℝ))
78	77	adantrd 491	. . . . . . . . . . . 12 ⊢ (𝐴 ⊆ ℝ → ((𝑧 ∈ 𝐴 ∧ 𝑧 < -𝑣) → 𝑧 ∈ ℝ))
79	78	pm4.71rd 562	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ ℝ → ((𝑧 ∈ 𝐴 ∧ 𝑧 < -𝑣) ↔ (𝑧 ∈ ℝ ∧ (𝑧 ∈ 𝐴 ∧ 𝑧 < -𝑣))))
80	79	exbidv 1923	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ℝ → (∃𝑧(𝑧 ∈ 𝐴 ∧ 𝑧 < -𝑣) ↔ ∃𝑧(𝑧 ∈ ℝ ∧ (𝑧 ∈ 𝐴 ∧ 𝑧 < -𝑣))))
81		df-rex 3070	. . . . . . . . . 10 ⊢ (∃𝑧 ∈ 𝐴 𝑧 < -𝑣 ↔ ∃𝑧(𝑧 ∈ 𝐴 ∧ 𝑧 < -𝑣))
82		renegcl 11530	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ ℝ → -𝑡 ∈ ℝ)
83		infm3lem 12179	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ ℝ → ∃𝑡 ∈ ℝ 𝑧 = -𝑡)
84		eleq1 2820	. . . . . . . . . . . . 13 ⊢ (𝑧 = -𝑡 → (𝑧 ∈ 𝐴 ↔ -𝑡 ∈ 𝐴))
85		breq1 5151	. . . . . . . . . . . . 13 ⊢ (𝑧 = -𝑡 → (𝑧 < -𝑣 ↔ -𝑡 < -𝑣))
86	84, 85	anbi12d 630	. . . . . . . . . . . 12 ⊢ (𝑧 = -𝑡 → ((𝑧 ∈ 𝐴 ∧ 𝑧 < -𝑣) ↔ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣)))
87	82, 83, 86	rexxfr 5414	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ ℝ (𝑧 ∈ 𝐴 ∧ 𝑧 < -𝑣) ↔ ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣))
88		df-rex 3070	. . . . . . . . . . 11 ⊢ (∃𝑧 ∈ ℝ (𝑧 ∈ 𝐴 ∧ 𝑧 < -𝑣) ↔ ∃𝑧(𝑧 ∈ ℝ ∧ (𝑧 ∈ 𝐴 ∧ 𝑧 < -𝑣)))
89	87, 88	bitr3i 277	. . . . . . . . . 10 ⊢ (∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣) ↔ ∃𝑧(𝑧 ∈ ℝ ∧ (𝑧 ∈ 𝐴 ∧ 𝑧 < -𝑣)))
90	80, 81, 89	3bitr4g 314	. . . . . . . . 9 ⊢ (𝐴 ⊆ ℝ → (∃𝑧 ∈ 𝐴 𝑧 < -𝑣 ↔ ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣)))
91	90	imbi2d 340	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ → ((𝑥 < -𝑣 → ∃𝑧 ∈ 𝐴 𝑧 < -𝑣) ↔ (𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣))))
92	91	ralbidv 3176	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ → (∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑧 ∈ 𝐴 𝑧 < -𝑣) ↔ ∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣))))
93	76, 92	bitrid 283	. . . . . 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 630	. . . . 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 3177	. . . 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 5152	. . . . . . . . . 10 ⊢ (𝑥 = -𝑢 → (-𝑣 < 𝑥 ↔ -𝑣 < -𝑢))
97	96	notbid 318	. . . . . . . . 9 ⊢ (𝑥 = -𝑢 → (¬ -𝑣 < 𝑥 ↔ ¬ -𝑣 < -𝑢))
98	97	imbi2d 340	. . . . . . . 8 ⊢ (𝑥 = -𝑢 → ((-𝑣 ∈ 𝐴 → ¬ -𝑣 < 𝑥) ↔ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < -𝑢)))
99	98	ralbidv 3176	. . . . . . 7 ⊢ (𝑥 = -𝑢 → (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < 𝑥) ↔ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < -𝑢)))
100		breq1 5151	. . . . . . . . 9 ⊢ (𝑥 = -𝑢 → (𝑥 < -𝑣 ↔ -𝑢 < -𝑣))
101	100	imbi1d 341	. . . . . . . 8 ⊢ (𝑥 = -𝑢 → ((𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣)) ↔ (-𝑢 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣))))
102	101	ralbidv 3176	. . . . . . 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 630	. . . . . 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 5414	. . . . 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 349	. . . . . . . . . . 11 ⊢ ((𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} → ¬ 𝑢 < 𝑣) ↔ ((𝑣 ∈ ℝ ∧ -𝑣 ∈ 𝐴) → ¬ 𝑢 < 𝑣))
106		impexp 450	. . . . . . . . . . 11 ⊢ (((𝑣 ∈ ℝ ∧ -𝑣 ∈ 𝐴) → ¬ 𝑢 < 𝑣) ↔ (𝑣 ∈ ℝ → (-𝑣 ∈ 𝐴 → ¬ 𝑢 < 𝑣)))
107	105, 106	bitri 275	. . . . . . . . . 10 ⊢ ((𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} → ¬ 𝑢 < 𝑣) ↔ (𝑣 ∈ ℝ → (-𝑣 ∈ 𝐴 → ¬ 𝑢 < 𝑣)))
108	107	albii 1820	. . . . . . . . 9 ⊢ (∀𝑣(𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} → ¬ 𝑢 < 𝑣) ↔ ∀𝑣(𝑣 ∈ ℝ → (-𝑣 ∈ 𝐴 → ¬ 𝑢 < 𝑣)))
109		df-ral 3061	. . . . . . . . 9 ⊢ (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑢 < 𝑣 ↔ ∀𝑣(𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} → ¬ 𝑢 < 𝑣))
110		df-ral 3061	. . . . . . . . 9 ⊢ (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ 𝑢 < 𝑣) ↔ ∀𝑣(𝑣 ∈ ℝ → (-𝑣 ∈ 𝐴 → ¬ 𝑢 < 𝑣)))
111	108, 109, 110	3bitr4ri 304	. . . . . . . 8 ⊢ (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ 𝑢 < 𝑣) ↔ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑢 < 𝑣)
112		ltneg 11721	. . . . . . . . . . 11 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢 < 𝑣 ↔ -𝑣 < -𝑢))
113	112	notbid 318	. . . . . . . . . 10 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (¬ 𝑢 < 𝑣 ↔ ¬ -𝑣 < -𝑢))
114	113	imbi2d 340	. . . . . . . . 9 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((-𝑣 ∈ 𝐴 → ¬ 𝑢 < 𝑣) ↔ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < -𝑢)))
115	114	ralbidva 3174	. . . . . . . 8 ⊢ (𝑢 ∈ ℝ → (∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ 𝑢 < 𝑣) ↔ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < -𝑢)))
116	111, 115	bitr3id 285	. . . . . . 7 ⊢ (𝑢 ∈ ℝ → (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴} ¬ 𝑢 < 𝑣 ↔ ∀𝑣 ∈ ℝ (-𝑣 ∈ 𝐴 → ¬ -𝑣 < -𝑢)))
117		ltneg 11721	. . . . . . . . . 10 ⊢ ((𝑣 ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑣 < 𝑢 ↔ -𝑢 < -𝑣))
118	117	ancoms 458	. . . . . . . . 9 ⊢ ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑣 < 𝑢 ↔ -𝑢 < -𝑣))
119		negeq 11459	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝑡 → -𝑤 = -𝑡)
120	119	eleq1d 2817	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝑡 → (-𝑤 ∈ 𝐴 ↔ -𝑡 ∈ 𝐴))
121	120	rexrab 3692	. . . . . . . . . . 11 ⊢ (∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤 ∈ 𝐴}𝑣 < 𝑡 ↔ ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ 𝑣 < 𝑡))
122		ltneg 11721	. . . . . . . . . . . . 13 ⊢ ((𝑣 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑣 < 𝑡 ↔ -𝑡 < -𝑣))
123	122	anbi2d 628	. . . . . . . . . . . 12 ⊢ ((𝑣 ∈ ℝ ∧ 𝑡 ∈ ℝ) → ((-𝑡 ∈ 𝐴 ∧ 𝑣 < 𝑡) ↔ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣)))
124	123	rexbidva 3175	. . . . . . . . . . 11 ⊢ (𝑣 ∈ ℝ → (∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ 𝑣 < 𝑡) ↔ ∃𝑡 ∈ ℝ (-𝑡 ∈ 𝐴 ∧ -𝑡 < -𝑣)))
125	121, 124	bitrid 283	. . . . . . . . . 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 344	. . . . . . . 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 3174	. . . . . . 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 630	. . . . . 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 3091	. . . . 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 278	. . . 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	bitrdi 287	. . 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 259	. 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 1115	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 205   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540  ∃wex 1780   ∈ wcel 2105   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  {crab 3431   ⊆ wss 3948  ∅c0 4322   class class class wbr 5148  ℝcr 11115   < clt 11255   ≤ cle 11256  -cneg 11452

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729  ax-resscn 11173  ax-1cn 11174  ax-icn 11175  ax-addcl 11176  ax-addrcl 11177  ax-mulcl 11178  ax-mulrcl 11179  ax-mulcom 11180  ax-addass 11181  ax-mulass 11182  ax-distr 11183  ax-i2m1 11184  ax-1ne0 11185  ax-1rid 11186  ax-rnegex 11187  ax-rrecex 11188  ax-cnre 11189  ax-pre-lttri 11190  ax-pre-lttrn 11191  ax-pre-ltadd 11192  ax-pre-mulgt0 11193  ax-pre-sup 11194

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-po 5588  df-so 5589  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-riota 7368  df-ov 7415  df-oprab 7416  df-mpo 7417  df-er 8709  df-en 8946  df-dom 8947  df-sdom 8948  df-pnf 11257  df-mnf 11258  df-xr 11259  df-ltxr 11260  df-le 11261  df-sub 11453  df-neg 11454

This theorem is referenced by:  infrecl  12203  infrenegsup  12204  infregelb  12205  infrelb  12206  xrinfmsslem  13294  gtinf  35668  infrglb  44765