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Theorem infm3 11341
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 11339.) (Contributed by NM, 14-Jun-2005.)
Assertion
Ref Expression
infm3 ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑥𝑦) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 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.
StepHypRef Expression
1 ssel 3815 . . . . . . . . 9 (𝐴 ⊆ ℝ → (𝑣𝐴𝑣 ∈ ℝ))
21pm4.71rd 558 . . . . . . . 8 (𝐴 ⊆ ℝ → (𝑣𝐴 ↔ (𝑣 ∈ ℝ ∧ 𝑣𝐴)))
32exbidv 1964 . . . . . . 7 (𝐴 ⊆ ℝ → (∃𝑣 𝑣𝐴 ↔ ∃𝑣(𝑣 ∈ ℝ ∧ 𝑣𝐴)))
4 df-rex 3096 . . . . . . . 8 (∃𝑣 ∈ ℝ 𝑣𝐴 ↔ ∃𝑣(𝑣 ∈ ℝ ∧ 𝑣𝐴))
5 renegcl 10688 . . . . . . . . 9 (𝑤 ∈ ℝ → -𝑤 ∈ ℝ)
6 infm3lem 11340 . . . . . . . . 9 (𝑣 ∈ ℝ → ∃𝑤 ∈ ℝ 𝑣 = -𝑤)
7 eleq1 2847 . . . . . . . . 9 (𝑣 = -𝑤 → (𝑣𝐴 ↔ -𝑤𝐴))
85, 6, 7rexxfr 5130 . . . . . . . 8 (∃𝑣 ∈ ℝ 𝑣𝐴 ↔ ∃𝑤 ∈ ℝ -𝑤𝐴)
94, 8bitr3i 269 . . . . . . 7 (∃𝑣(𝑣 ∈ ℝ ∧ 𝑣𝐴) ↔ ∃𝑤 ∈ ℝ -𝑤𝐴)
103, 9syl6bb 279 . . . . . 6 (𝐴 ⊆ ℝ → (∃𝑣 𝑣𝐴 ↔ ∃𝑤 ∈ ℝ -𝑤𝐴))
11 n0 4159 . . . . . 6 (𝐴 ≠ ∅ ↔ ∃𝑣 𝑣𝐴)
12 rabn0 4188 . . . . . 6 ({𝑤 ∈ ℝ ∣ -𝑤𝐴} ≠ ∅ ↔ ∃𝑤 ∈ ℝ -𝑤𝐴)
1310, 11, 123bitr4g 306 . . . . 5 (𝐴 ⊆ ℝ → (𝐴 ≠ ∅ ↔ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ≠ ∅))
14 ssel 3815 . . . . . . . . . . . 12 (𝐴 ⊆ ℝ → (𝑦𝐴𝑦 ∈ ℝ))
1514pm4.71rd 558 . . . . . . . . . . 11 (𝐴 ⊆ ℝ → (𝑦𝐴 ↔ (𝑦 ∈ ℝ ∧ 𝑦𝐴)))
1615imbi1d 333 . . . . . . . . . 10 (𝐴 ⊆ ℝ → ((𝑦𝐴𝑥𝑦) ↔ ((𝑦 ∈ ℝ ∧ 𝑦𝐴) → 𝑥𝑦)))
17 impexp 443 . . . . . . . . . 10 (((𝑦 ∈ ℝ ∧ 𝑦𝐴) → 𝑥𝑦) ↔ (𝑦 ∈ ℝ → (𝑦𝐴𝑥𝑦)))
1816, 17syl6bb 279 . . . . . . . . 9 (𝐴 ⊆ ℝ → ((𝑦𝐴𝑥𝑦) ↔ (𝑦 ∈ ℝ → (𝑦𝐴𝑥𝑦))))
1918albidv 1963 . . . . . . . 8 (𝐴 ⊆ ℝ → (∀𝑦(𝑦𝐴𝑥𝑦) ↔ ∀𝑦(𝑦 ∈ ℝ → (𝑦𝐴𝑥𝑦))))
20 df-ral 3095 . . . . . . . 8 (∀𝑦𝐴 𝑥𝑦 ↔ ∀𝑦(𝑦𝐴𝑥𝑦))
21 renegcl 10688 . . . . . . . . . 10 (𝑣 ∈ ℝ → -𝑣 ∈ ℝ)
22 infm3lem 11340 . . . . . . . . . 10 (𝑦 ∈ ℝ → ∃𝑣 ∈ ℝ 𝑦 = -𝑣)
23 eleq1 2847 . . . . . . . . . . 11 (𝑦 = -𝑣 → (𝑦𝐴 ↔ -𝑣𝐴))
24 breq2 4892 . . . . . . . . . . 11 (𝑦 = -𝑣 → (𝑥𝑦𝑥 ≤ -𝑣))
2523, 24imbi12d 336 . . . . . . . . . 10 (𝑦 = -𝑣 → ((𝑦𝐴𝑥𝑦) ↔ (-𝑣𝐴𝑥 ≤ -𝑣)))
2621, 22, 25ralxfr 5128 . . . . . . . . 9 (∀𝑦 ∈ ℝ (𝑦𝐴𝑥𝑦) ↔ ∀𝑣 ∈ ℝ (-𝑣𝐴𝑥 ≤ -𝑣))
27 df-ral 3095 . . . . . . . . 9 (∀𝑦 ∈ ℝ (𝑦𝐴𝑥𝑦) ↔ ∀𝑦(𝑦 ∈ ℝ → (𝑦𝐴𝑥𝑦)))
2826, 27bitr3i 269 . . . . . . . 8 (∀𝑣 ∈ ℝ (-𝑣𝐴𝑥 ≤ -𝑣) ↔ ∀𝑦(𝑦 ∈ ℝ → (𝑦𝐴𝑥𝑦)))
2919, 20, 283bitr4g 306 . . . . . . 7 (𝐴 ⊆ ℝ → (∀𝑦𝐴 𝑥𝑦 ↔ ∀𝑣 ∈ ℝ (-𝑣𝐴𝑥 ≤ -𝑣)))
3029rexbidv 3237 . . . . . 6 (𝐴 ⊆ ℝ → (∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑥𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑣 ∈ ℝ (-𝑣𝐴𝑥 ≤ -𝑣)))
31 renegcl 10688 . . . . . . . 8 (𝑢 ∈ ℝ → -𝑢 ∈ ℝ)
32 infm3lem 11340 . . . . . . . 8 (𝑥 ∈ ℝ → ∃𝑢 ∈ ℝ 𝑥 = -𝑢)
33 breq1 4891 . . . . . . . . . 10 (𝑥 = -𝑢 → (𝑥 ≤ -𝑣 ↔ -𝑢 ≤ -𝑣))
3433imbi2d 332 . . . . . . . . 9 (𝑥 = -𝑢 → ((-𝑣𝐴𝑥 ≤ -𝑣) ↔ (-𝑣𝐴 → -𝑢 ≤ -𝑣)))
3534ralbidv 3168 . . . . . . . 8 (𝑥 = -𝑢 → (∀𝑣 ∈ ℝ (-𝑣𝐴𝑥 ≤ -𝑣) ↔ ∀𝑣 ∈ ℝ (-𝑣𝐴 → -𝑢 ≤ -𝑣)))
3631, 32, 35rexxfr 5130 . . . . . . 7 (∃𝑥 ∈ ℝ ∀𝑣 ∈ ℝ (-𝑣𝐴𝑥 ≤ -𝑣) ↔ ∃𝑢 ∈ ℝ ∀𝑣 ∈ ℝ (-𝑣𝐴 → -𝑢 ≤ -𝑣))
37 negeq 10616 . . . . . . . . . . . . . . 15 (𝑤 = 𝑣 → -𝑤 = -𝑣)
3837eleq1d 2844 . . . . . . . . . . . . . 14 (𝑤 = 𝑣 → (-𝑤𝐴 ↔ -𝑣𝐴))
3938elrab 3572 . . . . . . . . . . . . 13 (𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ↔ (𝑣 ∈ ℝ ∧ -𝑣𝐴))
4039imbi1i 341 . . . . . . . . . . . 12 ((𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → 𝑣𝑢) ↔ ((𝑣 ∈ ℝ ∧ -𝑣𝐴) → 𝑣𝑢))
41 impexp 443 . . . . . . . . . . . 12 (((𝑣 ∈ ℝ ∧ -𝑣𝐴) → 𝑣𝑢) ↔ (𝑣 ∈ ℝ → (-𝑣𝐴𝑣𝑢)))
4240, 41bitri 267 . . . . . . . . . . 11 ((𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → 𝑣𝑢) ↔ (𝑣 ∈ ℝ → (-𝑣𝐴𝑣𝑢)))
4342albii 1863 . . . . . . . . . 10 (∀𝑣(𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → 𝑣𝑢) ↔ ∀𝑣(𝑣 ∈ ℝ → (-𝑣𝐴𝑣𝑢)))
44 df-ral 3095 . . . . . . . . . 10 (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣𝑢 ↔ ∀𝑣(𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → 𝑣𝑢))
45 df-ral 3095 . . . . . . . . . 10 (∀𝑣 ∈ ℝ (-𝑣𝐴𝑣𝑢) ↔ ∀𝑣(𝑣 ∈ ℝ → (-𝑣𝐴𝑣𝑢)))
4643, 44, 453bitr4ri 296 . . . . . . . . 9 (∀𝑣 ∈ ℝ (-𝑣𝐴𝑣𝑢) ↔ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣𝑢)
47 leneg 10881 . . . . . . . . . . . 12 ((𝑣 ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑣𝑢 ↔ -𝑢 ≤ -𝑣))
4847ancoms 452 . . . . . . . . . . 11 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑣𝑢 ↔ -𝑢 ≤ -𝑣))
4948imbi2d 332 . . . . . . . . . 10 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((-𝑣𝐴𝑣𝑢) ↔ (-𝑣𝐴 → -𝑢 ≤ -𝑣)))
5049ralbidva 3167 . . . . . . . . 9 (𝑢 ∈ ℝ → (∀𝑣 ∈ ℝ (-𝑣𝐴𝑣𝑢) ↔ ∀𝑣 ∈ ℝ (-𝑣𝐴 → -𝑢 ≤ -𝑣)))
5146, 50syl5bbr 277 . . . . . . . 8 (𝑢 ∈ ℝ → (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣𝑢 ↔ ∀𝑣 ∈ ℝ (-𝑣𝐴 → -𝑢 ≤ -𝑣)))
5251rexbiia 3223 . . . . . . 7 (∃𝑢 ∈ ℝ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣𝑢 ↔ ∃𝑢 ∈ ℝ ∀𝑣 ∈ ℝ (-𝑣𝐴 → -𝑢 ≤ -𝑣))
5336, 52bitr4i 270 . . . . . 6 (∃𝑥 ∈ ℝ ∀𝑣 ∈ ℝ (-𝑣𝐴𝑥 ≤ -𝑣) ↔ ∃𝑢 ∈ ℝ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣𝑢)
5430, 53syl6bb 279 . . . . 5 (𝐴 ⊆ ℝ → (∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑥𝑦 ↔ ∃𝑢 ∈ ℝ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣𝑢))
5513, 54anbi12d 624 . . . 4 (𝐴 ⊆ ℝ → ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑥𝑦) ↔ ({𝑤 ∈ ℝ ∣ -𝑤𝐴} ≠ ∅ ∧ ∃𝑢 ∈ ℝ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣𝑢)))
56 ssrab2 3908 . . . . 5 {𝑤 ∈ ℝ ∣ -𝑤𝐴} ⊆ ℝ
57 sup3 11339 . . . . 5 (({𝑤 ∈ ℝ ∣ -𝑤𝐴} ⊆ ℝ ∧ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ≠ ∅ ∧ ∃𝑢 ∈ ℝ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣𝑢) → ∃𝑢 ∈ ℝ (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑢 < 𝑣 ∧ ∀𝑣 ∈ ℝ (𝑣 < 𝑢 → ∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣 < 𝑡)))
5856, 57mp3an1 1521 . . . 4 (({𝑤 ∈ ℝ ∣ -𝑤𝐴} ≠ ∅ ∧ ∃𝑢 ∈ ℝ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣𝑢) → ∃𝑢 ∈ ℝ (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑢 < 𝑣 ∧ ∀𝑣 ∈ ℝ (𝑣 < 𝑢 → ∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣 < 𝑡)))
5955, 58syl6bi 245 . . 3 (𝐴 ⊆ ℝ → ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑥𝑦) → ∃𝑢 ∈ ℝ (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑢 < 𝑣 ∧ ∀𝑣 ∈ ℝ (𝑣 < 𝑢 → ∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣 < 𝑡))))
6015imbi1d 333 . . . . . . . . 9 (𝐴 ⊆ ℝ → ((𝑦𝐴 → ¬ 𝑦 < 𝑥) ↔ ((𝑦 ∈ ℝ ∧ 𝑦𝐴) → ¬ 𝑦 < 𝑥)))
61 impexp 443 . . . . . . . . 9 (((𝑦 ∈ ℝ ∧ 𝑦𝐴) → ¬ 𝑦 < 𝑥) ↔ (𝑦 ∈ ℝ → (𝑦𝐴 → ¬ 𝑦 < 𝑥)))
6260, 61syl6bb 279 . . . . . . . 8 (𝐴 ⊆ ℝ → ((𝑦𝐴 → ¬ 𝑦 < 𝑥) ↔ (𝑦 ∈ ℝ → (𝑦𝐴 → ¬ 𝑦 < 𝑥))))
6362albidv 1963 . . . . . . 7 (𝐴 ⊆ ℝ → (∀𝑦(𝑦𝐴 → ¬ 𝑦 < 𝑥) ↔ ∀𝑦(𝑦 ∈ ℝ → (𝑦𝐴 → ¬ 𝑦 < 𝑥))))
64 df-ral 3095 . . . . . . 7 (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦(𝑦𝐴 → ¬ 𝑦 < 𝑥))
65 breq1 4891 . . . . . . . . . . 11 (𝑦 = -𝑣 → (𝑦 < 𝑥 ↔ -𝑣 < 𝑥))
6665notbid 310 . . . . . . . . . 10 (𝑦 = -𝑣 → (¬ 𝑦 < 𝑥 ↔ ¬ -𝑣 < 𝑥))
6723, 66imbi12d 336 . . . . . . . . 9 (𝑦 = -𝑣 → ((𝑦𝐴 → ¬ 𝑦 < 𝑥) ↔ (-𝑣𝐴 → ¬ -𝑣 < 𝑥)))
6821, 22, 67ralxfr 5128 . . . . . . . 8 (∀𝑦 ∈ ℝ (𝑦𝐴 → ¬ 𝑦 < 𝑥) ↔ ∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ -𝑣 < 𝑥))
69 df-ral 3095 . . . . . . . 8 (∀𝑦 ∈ ℝ (𝑦𝐴 → ¬ 𝑦 < 𝑥) ↔ ∀𝑦(𝑦 ∈ ℝ → (𝑦𝐴 → ¬ 𝑦 < 𝑥)))
7068, 69bitr3i 269 . . . . . . 7 (∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ -𝑣 < 𝑥) ↔ ∀𝑦(𝑦 ∈ ℝ → (𝑦𝐴 → ¬ 𝑦 < 𝑥)))
7163, 64, 703bitr4g 306 . . . . . 6 (𝐴 ⊆ ℝ → (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ -𝑣 < 𝑥)))
72 breq2 4892 . . . . . . . . 9 (𝑦 = -𝑣 → (𝑥 < 𝑦𝑥 < -𝑣))
73 breq2 4892 . . . . . . . . . 10 (𝑦 = -𝑣 → (𝑧 < 𝑦𝑧 < -𝑣))
7473rexbidv 3237 . . . . . . . . 9 (𝑦 = -𝑣 → (∃𝑧𝐴 𝑧 < 𝑦 ↔ ∃𝑧𝐴 𝑧 < -𝑣))
7572, 74imbi12d 336 . . . . . . . 8 (𝑦 = -𝑣 → ((𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ (𝑥 < -𝑣 → ∃𝑧𝐴 𝑧 < -𝑣)))
7621, 22, 75ralxfr 5128 . . . . . . 7 (∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ ∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑧𝐴 𝑧 < -𝑣))
77 ssel 3815 . . . . . . . . . . . . 13 (𝐴 ⊆ ℝ → (𝑧𝐴𝑧 ∈ ℝ))
7877adantrd 487 . . . . . . . . . . . 12 (𝐴 ⊆ ℝ → ((𝑧𝐴𝑧 < -𝑣) → 𝑧 ∈ ℝ))
7978pm4.71rd 558 . . . . . . . . . . 11 (𝐴 ⊆ ℝ → ((𝑧𝐴𝑧 < -𝑣) ↔ (𝑧 ∈ ℝ ∧ (𝑧𝐴𝑧 < -𝑣))))
8079exbidv 1964 . . . . . . . . . 10 (𝐴 ⊆ ℝ → (∃𝑧(𝑧𝐴𝑧 < -𝑣) ↔ ∃𝑧(𝑧 ∈ ℝ ∧ (𝑧𝐴𝑧 < -𝑣))))
81 df-rex 3096 . . . . . . . . . 10 (∃𝑧𝐴 𝑧 < -𝑣 ↔ ∃𝑧(𝑧𝐴𝑧 < -𝑣))
82 renegcl 10688 . . . . . . . . . . . 12 (𝑡 ∈ ℝ → -𝑡 ∈ ℝ)
83 infm3lem 11340 . . . . . . . . . . . 12 (𝑧 ∈ ℝ → ∃𝑡 ∈ ℝ 𝑧 = -𝑡)
84 eleq1 2847 . . . . . . . . . . . . 13 (𝑧 = -𝑡 → (𝑧𝐴 ↔ -𝑡𝐴))
85 breq1 4891 . . . . . . . . . . . . 13 (𝑧 = -𝑡 → (𝑧 < -𝑣 ↔ -𝑡 < -𝑣))
8684, 85anbi12d 624 . . . . . . . . . . . 12 (𝑧 = -𝑡 → ((𝑧𝐴𝑧 < -𝑣) ↔ (-𝑡𝐴 ∧ -𝑡 < -𝑣)))
8782, 83, 86rexxfr 5130 . . . . . . . . . . 11 (∃𝑧 ∈ ℝ (𝑧𝐴𝑧 < -𝑣) ↔ ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣))
88 df-rex 3096 . . . . . . . . . . 11 (∃𝑧 ∈ ℝ (𝑧𝐴𝑧 < -𝑣) ↔ ∃𝑧(𝑧 ∈ ℝ ∧ (𝑧𝐴𝑧 < -𝑣)))
8987, 88bitr3i 269 . . . . . . . . . 10 (∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣) ↔ ∃𝑧(𝑧 ∈ ℝ ∧ (𝑧𝐴𝑧 < -𝑣)))
9080, 81, 893bitr4g 306 . . . . . . . . 9 (𝐴 ⊆ ℝ → (∃𝑧𝐴 𝑧 < -𝑣 ↔ ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣)))
9190imbi2d 332 . . . . . . . 8 (𝐴 ⊆ ℝ → ((𝑥 < -𝑣 → ∃𝑧𝐴 𝑧 < -𝑣) ↔ (𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣))))
9291ralbidv 3168 . . . . . . 7 (𝐴 ⊆ ℝ → (∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑧𝐴 𝑧 < -𝑣) ↔ ∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣))))
9376, 92syl5bb 275 . . . . . 6 (𝐴 ⊆ ℝ → (∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ ∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣))))
9471, 93anbi12d 624 . . . . 5 (𝐴 ⊆ ℝ → ((∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) ↔ (∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ -𝑣 < 𝑥) ∧ ∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣)))))
9594rexbidv 3237 . . . 4 (𝐴 ⊆ ℝ → (∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) ↔ ∃𝑥 ∈ ℝ (∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ -𝑣 < 𝑥) ∧ ∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣)))))
96 breq2 4892 . . . . . . . . . 10 (𝑥 = -𝑢 → (-𝑣 < 𝑥 ↔ -𝑣 < -𝑢))
9796notbid 310 . . . . . . . . 9 (𝑥 = -𝑢 → (¬ -𝑣 < 𝑥 ↔ ¬ -𝑣 < -𝑢))
9897imbi2d 332 . . . . . . . 8 (𝑥 = -𝑢 → ((-𝑣𝐴 → ¬ -𝑣 < 𝑥) ↔ (-𝑣𝐴 → ¬ -𝑣 < -𝑢)))
9998ralbidv 3168 . . . . . . 7 (𝑥 = -𝑢 → (∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ -𝑣 < 𝑥) ↔ ∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ -𝑣 < -𝑢)))
100 breq1 4891 . . . . . . . . 9 (𝑥 = -𝑢 → (𝑥 < -𝑣 ↔ -𝑢 < -𝑣))
101100imbi1d 333 . . . . . . . 8 (𝑥 = -𝑢 → ((𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣)) ↔ (-𝑢 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣))))
102101ralbidv 3168 . . . . . . 7 (𝑥 = -𝑢 → (∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣)) ↔ ∀𝑣 ∈ ℝ (-𝑢 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣))))
10399, 102anbi12d 624 . . . . . 6 (𝑥 = -𝑢 → ((∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ -𝑣 < 𝑥) ∧ ∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣))) ↔ (∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ -𝑣 < -𝑢) ∧ ∀𝑣 ∈ ℝ (-𝑢 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣)))))
10431, 32, 103rexxfr 5130 . . . . 5 (∃𝑥 ∈ ℝ (∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ -𝑣 < 𝑥) ∧ ∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣))) ↔ ∃𝑢 ∈ ℝ (∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ -𝑣 < -𝑢) ∧ ∀𝑣 ∈ ℝ (-𝑢 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣))))
10539imbi1i 341 . . . . . . . . . . 11 ((𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → ¬ 𝑢 < 𝑣) ↔ ((𝑣 ∈ ℝ ∧ -𝑣𝐴) → ¬ 𝑢 < 𝑣))
106 impexp 443 . . . . . . . . . . 11 (((𝑣 ∈ ℝ ∧ -𝑣𝐴) → ¬ 𝑢 < 𝑣) ↔ (𝑣 ∈ ℝ → (-𝑣𝐴 → ¬ 𝑢 < 𝑣)))
107105, 106bitri 267 . . . . . . . . . 10 ((𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → ¬ 𝑢 < 𝑣) ↔ (𝑣 ∈ ℝ → (-𝑣𝐴 → ¬ 𝑢 < 𝑣)))
108107albii 1863 . . . . . . . . 9 (∀𝑣(𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → ¬ 𝑢 < 𝑣) ↔ ∀𝑣(𝑣 ∈ ℝ → (-𝑣𝐴 → ¬ 𝑢 < 𝑣)))
109 df-ral 3095 . . . . . . . . 9 (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑢 < 𝑣 ↔ ∀𝑣(𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → ¬ 𝑢 < 𝑣))
110 df-ral 3095 . . . . . . . . 9 (∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ 𝑢 < 𝑣) ↔ ∀𝑣(𝑣 ∈ ℝ → (-𝑣𝐴 → ¬ 𝑢 < 𝑣)))
111108, 109, 1103bitr4ri 296 . . . . . . . 8 (∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ 𝑢 < 𝑣) ↔ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑢 < 𝑣)
112 ltneg 10878 . . . . . . . . . . 11 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢 < 𝑣 ↔ -𝑣 < -𝑢))
113112notbid 310 . . . . . . . . . 10 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (¬ 𝑢 < 𝑣 ↔ ¬ -𝑣 < -𝑢))
114113imbi2d 332 . . . . . . . . 9 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((-𝑣𝐴 → ¬ 𝑢 < 𝑣) ↔ (-𝑣𝐴 → ¬ -𝑣 < -𝑢)))
115114ralbidva 3167 . . . . . . . 8 (𝑢 ∈ ℝ → (∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ 𝑢 < 𝑣) ↔ ∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ -𝑣 < -𝑢)))
116111, 115syl5bbr 277 . . . . . . 7 (𝑢 ∈ ℝ → (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑢 < 𝑣 ↔ ∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ -𝑣 < -𝑢)))
117 ltneg 10878 . . . . . . . . . 10 ((𝑣 ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑣 < 𝑢 ↔ -𝑢 < -𝑣))
118117ancoms 452 . . . . . . . . 9 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑣 < 𝑢 ↔ -𝑢 < -𝑣))
119 negeq 10616 . . . . . . . . . . . . 13 (𝑤 = 𝑡 → -𝑤 = -𝑡)
120119eleq1d 2844 . . . . . . . . . . . 12 (𝑤 = 𝑡 → (-𝑤𝐴 ↔ -𝑡𝐴))
121120rexrab 3580 . . . . . . . . . . 11 (∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣 < 𝑡 ↔ ∃𝑡 ∈ ℝ (-𝑡𝐴𝑣 < 𝑡))
122 ltneg 10878 . . . . . . . . . . . . 13 ((𝑣 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑣 < 𝑡 ↔ -𝑡 < -𝑣))
123122anbi2d 622 . . . . . . . . . . . 12 ((𝑣 ∈ ℝ ∧ 𝑡 ∈ ℝ) → ((-𝑡𝐴𝑣 < 𝑡) ↔ (-𝑡𝐴 ∧ -𝑡 < -𝑣)))
124123rexbidva 3234 . . . . . . . . . . 11 (𝑣 ∈ ℝ → (∃𝑡 ∈ ℝ (-𝑡𝐴𝑣 < 𝑡) ↔ ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣)))
125121, 124syl5bb 275 . . . . . . . . . 10 (𝑣 ∈ ℝ → (∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣 < 𝑡 ↔ ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣)))
126125adantl 475 . . . . . . . . 9 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣 < 𝑡 ↔ ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣)))
127118, 126imbi12d 336 . . . . . . . 8 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((𝑣 < 𝑢 → ∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣 < 𝑡) ↔ (-𝑢 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣))))
128127ralbidva 3167 . . . . . . 7 (𝑢 ∈ ℝ → (∀𝑣 ∈ ℝ (𝑣 < 𝑢 → ∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣 < 𝑡) ↔ ∀𝑣 ∈ ℝ (-𝑢 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣))))
129116, 128anbi12d 624 . . . . . 6 (𝑢 ∈ ℝ → ((∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑢 < 𝑣 ∧ ∀𝑣 ∈ ℝ (𝑣 < 𝑢 → ∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣 < 𝑡)) ↔ (∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ -𝑣 < -𝑢) ∧ ∀𝑣 ∈ ℝ (-𝑢 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣)))))
130129rexbiia 3223 . . . . 5 (∃𝑢 ∈ ℝ (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑢 < 𝑣 ∧ ∀𝑣 ∈ ℝ (𝑣 < 𝑢 → ∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣 < 𝑡)) ↔ ∃𝑢 ∈ ℝ (∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ -𝑣 < -𝑢) ∧ ∀𝑣 ∈ ℝ (-𝑢 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣))))
131104, 130bitr4i 270 . . . 4 (∃𝑥 ∈ ℝ (∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ -𝑣 < 𝑥) ∧ ∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣))) ↔ ∃𝑢 ∈ ℝ (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑢 < 𝑣 ∧ ∀𝑣 ∈ ℝ (𝑣 < 𝑢 → ∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣 < 𝑡)))
13295, 131syl6bb 279 . . 3 (𝐴 ⊆ ℝ → (∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) ↔ ∃𝑢 ∈ ℝ (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑢 < 𝑣 ∧ ∀𝑣 ∈ ℝ (𝑣 < 𝑢 → ∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣 < 𝑡))))
13359, 132sylibrd 251 . 2 (𝐴 ⊆ ℝ → ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑥𝑦) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))))
1341333impib 1105 1 ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑥𝑦) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  w3a 1071  wal 1599   = wceq 1601  wex 1823  wcel 2107  wne 2969  wral 3090  wrex 3091  {crab 3094  wss 3792  c0 4141   class class class wbr 4888  cr 10273   < clt 10413  cle 10414  -cneg 10609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5019  ax-nul 5027  ax-pow 5079  ax-pr 5140  ax-un 7228  ax-resscn 10331  ax-1cn 10332  ax-icn 10333  ax-addcl 10334  ax-addrcl 10335  ax-mulcl 10336  ax-mulrcl 10337  ax-mulcom 10338  ax-addass 10339  ax-mulass 10340  ax-distr 10341  ax-i2m1 10342  ax-1ne0 10343  ax-1rid 10344  ax-rnegex 10345  ax-rrecex 10346  ax-cnre 10347  ax-pre-lttri 10348  ax-pre-lttrn 10349  ax-pre-ltadd 10350  ax-pre-mulgt0 10351  ax-pre-sup 10352
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4674  df-br 4889  df-opab 4951  df-mpt 4968  df-id 5263  df-po 5276  df-so 5277  df-xp 5363  df-rel 5364  df-cnv 5365  df-co 5366  df-dm 5367  df-rn 5368  df-res 5369  df-ima 5370  df-iota 6101  df-fun 6139  df-fn 6140  df-f 6141  df-f1 6142  df-fo 6143  df-f1o 6144  df-fv 6145  df-riota 6885  df-ov 6927  df-oprab 6928  df-mpt2 6929  df-er 8028  df-en 8244  df-dom 8245  df-sdom 8246  df-pnf 10415  df-mnf 10416  df-xr 10417  df-ltxr 10418  df-le 10419  df-sub 10610  df-neg 10611
This theorem is referenced by:  infrecl  11364  infrenegsup  11365  infregelb  11366  infrelb  11367  xrinfmsslem  12455  gtinf  32910  infrglb  40744
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