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Theorem infm3 11274
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 11272.) (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 3803 . . . . . . . . 9 (𝐴 ⊆ ℝ → (𝑣𝐴𝑣 ∈ ℝ))
21pm4.71rd 554 . . . . . . . 8 (𝐴 ⊆ ℝ → (𝑣𝐴 ↔ (𝑣 ∈ ℝ ∧ 𝑣𝐴)))
32exbidv 2012 . . . . . . 7 (𝐴 ⊆ ℝ → (∃𝑣 𝑣𝐴 ↔ ∃𝑣(𝑣 ∈ ℝ ∧ 𝑣𝐴)))
4 df-rex 3113 . . . . . . . 8 (∃𝑣 ∈ ℝ 𝑣𝐴 ↔ ∃𝑣(𝑣 ∈ ℝ ∧ 𝑣𝐴))
5 renegcl 10636 . . . . . . . . 9 (𝑤 ∈ ℝ → -𝑤 ∈ ℝ)
6 infm3lem 11273 . . . . . . . . 9 (𝑣 ∈ ℝ → ∃𝑤 ∈ ℝ 𝑣 = -𝑤)
7 eleq1 2884 . . . . . . . . 9 (𝑣 = -𝑤 → (𝑣𝐴 ↔ -𝑤𝐴))
85, 6, 7rexxfr 5096 . . . . . . . 8 (∃𝑣 ∈ ℝ 𝑣𝐴 ↔ ∃𝑤 ∈ ℝ -𝑤𝐴)
94, 8bitr3i 268 . . . . . . 7 (∃𝑣(𝑣 ∈ ℝ ∧ 𝑣𝐴) ↔ ∃𝑤 ∈ ℝ -𝑤𝐴)
103, 9syl6bb 278 . . . . . 6 (𝐴 ⊆ ℝ → (∃𝑣 𝑣𝐴 ↔ ∃𝑤 ∈ ℝ -𝑤𝐴))
11 n0 4143 . . . . . 6 (𝐴 ≠ ∅ ↔ ∃𝑣 𝑣𝐴)
12 rabn0 4169 . . . . . 6 ({𝑤 ∈ ℝ ∣ -𝑤𝐴} ≠ ∅ ↔ ∃𝑤 ∈ ℝ -𝑤𝐴)
1310, 11, 123bitr4g 305 . . . . 5 (𝐴 ⊆ ℝ → (𝐴 ≠ ∅ ↔ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ≠ ∅))
14 ssel 3803 . . . . . . . . . . . 12 (𝐴 ⊆ ℝ → (𝑦𝐴𝑦 ∈ ℝ))
1514pm4.71rd 554 . . . . . . . . . . 11 (𝐴 ⊆ ℝ → (𝑦𝐴 ↔ (𝑦 ∈ ℝ ∧ 𝑦𝐴)))
1615imbi1d 332 . . . . . . . . . 10 (𝐴 ⊆ ℝ → ((𝑦𝐴𝑥𝑦) ↔ ((𝑦 ∈ ℝ ∧ 𝑦𝐴) → 𝑥𝑦)))
17 impexp 439 . . . . . . . . . 10 (((𝑦 ∈ ℝ ∧ 𝑦𝐴) → 𝑥𝑦) ↔ (𝑦 ∈ ℝ → (𝑦𝐴𝑥𝑦)))
1816, 17syl6bb 278 . . . . . . . . 9 (𝐴 ⊆ ℝ → ((𝑦𝐴𝑥𝑦) ↔ (𝑦 ∈ ℝ → (𝑦𝐴𝑥𝑦))))
1918albidv 2011 . . . . . . . 8 (𝐴 ⊆ ℝ → (∀𝑦(𝑦𝐴𝑥𝑦) ↔ ∀𝑦(𝑦 ∈ ℝ → (𝑦𝐴𝑥𝑦))))
20 df-ral 3112 . . . . . . . 8 (∀𝑦𝐴 𝑥𝑦 ↔ ∀𝑦(𝑦𝐴𝑥𝑦))
21 renegcl 10636 . . . . . . . . . 10 (𝑣 ∈ ℝ → -𝑣 ∈ ℝ)
22 infm3lem 11273 . . . . . . . . . 10 (𝑦 ∈ ℝ → ∃𝑣 ∈ ℝ 𝑦 = -𝑣)
23 eleq1 2884 . . . . . . . . . . 11 (𝑦 = -𝑣 → (𝑦𝐴 ↔ -𝑣𝐴))
24 breq2 4859 . . . . . . . . . . 11 (𝑦 = -𝑣 → (𝑥𝑦𝑥 ≤ -𝑣))
2523, 24imbi12d 335 . . . . . . . . . 10 (𝑦 = -𝑣 → ((𝑦𝐴𝑥𝑦) ↔ (-𝑣𝐴𝑥 ≤ -𝑣)))
2621, 22, 25ralxfr 5094 . . . . . . . . 9 (∀𝑦 ∈ ℝ (𝑦𝐴𝑥𝑦) ↔ ∀𝑣 ∈ ℝ (-𝑣𝐴𝑥 ≤ -𝑣))
27 df-ral 3112 . . . . . . . . 9 (∀𝑦 ∈ ℝ (𝑦𝐴𝑥𝑦) ↔ ∀𝑦(𝑦 ∈ ℝ → (𝑦𝐴𝑥𝑦)))
2826, 27bitr3i 268 . . . . . . . 8 (∀𝑣 ∈ ℝ (-𝑣𝐴𝑥 ≤ -𝑣) ↔ ∀𝑦(𝑦 ∈ ℝ → (𝑦𝐴𝑥𝑦)))
2919, 20, 283bitr4g 305 . . . . . . 7 (𝐴 ⊆ ℝ → (∀𝑦𝐴 𝑥𝑦 ↔ ∀𝑣 ∈ ℝ (-𝑣𝐴𝑥 ≤ -𝑣)))
3029rexbidv 3251 . . . . . 6 (𝐴 ⊆ ℝ → (∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑥𝑦 ↔ ∃𝑥 ∈ ℝ ∀𝑣 ∈ ℝ (-𝑣𝐴𝑥 ≤ -𝑣)))
31 renegcl 10636 . . . . . . . 8 (𝑢 ∈ ℝ → -𝑢 ∈ ℝ)
32 infm3lem 11273 . . . . . . . 8 (𝑥 ∈ ℝ → ∃𝑢 ∈ ℝ 𝑥 = -𝑢)
33 breq1 4858 . . . . . . . . . 10 (𝑥 = -𝑢 → (𝑥 ≤ -𝑣 ↔ -𝑢 ≤ -𝑣))
3433imbi2d 331 . . . . . . . . 9 (𝑥 = -𝑢 → ((-𝑣𝐴𝑥 ≤ -𝑣) ↔ (-𝑣𝐴 → -𝑢 ≤ -𝑣)))
3534ralbidv 3185 . . . . . . . 8 (𝑥 = -𝑢 → (∀𝑣 ∈ ℝ (-𝑣𝐴𝑥 ≤ -𝑣) ↔ ∀𝑣 ∈ ℝ (-𝑣𝐴 → -𝑢 ≤ -𝑣)))
3631, 32, 35rexxfr 5096 . . . . . . 7 (∃𝑥 ∈ ℝ ∀𝑣 ∈ ℝ (-𝑣𝐴𝑥 ≤ -𝑣) ↔ ∃𝑢 ∈ ℝ ∀𝑣 ∈ ℝ (-𝑣𝐴 → -𝑢 ≤ -𝑣))
37 negeq 10565 . . . . . . . . . . . . . . 15 (𝑤 = 𝑣 → -𝑤 = -𝑣)
3837eleq1d 2881 . . . . . . . . . . . . . 14 (𝑤 = 𝑣 → (-𝑤𝐴 ↔ -𝑣𝐴))
3938elrab 3570 . . . . . . . . . . . . 13 (𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ↔ (𝑣 ∈ ℝ ∧ -𝑣𝐴))
4039imbi1i 340 . . . . . . . . . . . 12 ((𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → 𝑣𝑢) ↔ ((𝑣 ∈ ℝ ∧ -𝑣𝐴) → 𝑣𝑢))
41 impexp 439 . . . . . . . . . . . 12 (((𝑣 ∈ ℝ ∧ -𝑣𝐴) → 𝑣𝑢) ↔ (𝑣 ∈ ℝ → (-𝑣𝐴𝑣𝑢)))
4240, 41bitri 266 . . . . . . . . . . 11 ((𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → 𝑣𝑢) ↔ (𝑣 ∈ ℝ → (-𝑣𝐴𝑣𝑢)))
4342albii 1904 . . . . . . . . . 10 (∀𝑣(𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → 𝑣𝑢) ↔ ∀𝑣(𝑣 ∈ ℝ → (-𝑣𝐴𝑣𝑢)))
44 df-ral 3112 . . . . . . . . . 10 (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣𝑢 ↔ ∀𝑣(𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → 𝑣𝑢))
45 df-ral 3112 . . . . . . . . . 10 (∀𝑣 ∈ ℝ (-𝑣𝐴𝑣𝑢) ↔ ∀𝑣(𝑣 ∈ ℝ → (-𝑣𝐴𝑣𝑢)))
4643, 44, 453bitr4ri 295 . . . . . . . . 9 (∀𝑣 ∈ ℝ (-𝑣𝐴𝑣𝑢) ↔ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣𝑢)
47 leneg 10823 . . . . . . . . . . . 12 ((𝑣 ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑣𝑢 ↔ -𝑢 ≤ -𝑣))
4847ancoms 448 . . . . . . . . . . 11 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑣𝑢 ↔ -𝑢 ≤ -𝑣))
4948imbi2d 331 . . . . . . . . . 10 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((-𝑣𝐴𝑣𝑢) ↔ (-𝑣𝐴 → -𝑢 ≤ -𝑣)))
5049ralbidva 3184 . . . . . . . . 9 (𝑢 ∈ ℝ → (∀𝑣 ∈ ℝ (-𝑣𝐴𝑣𝑢) ↔ ∀𝑣 ∈ ℝ (-𝑣𝐴 → -𝑢 ≤ -𝑣)))
5146, 50syl5bbr 276 . . . . . . . 8 (𝑢 ∈ ℝ → (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣𝑢 ↔ ∀𝑣 ∈ ℝ (-𝑣𝐴 → -𝑢 ≤ -𝑣)))
5251rexbiia 3239 . . . . . . 7 (∃𝑢 ∈ ℝ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣𝑢 ↔ ∃𝑢 ∈ ℝ ∀𝑣 ∈ ℝ (-𝑣𝐴 → -𝑢 ≤ -𝑣))
5336, 52bitr4i 269 . . . . . 6 (∃𝑥 ∈ ℝ ∀𝑣 ∈ ℝ (-𝑣𝐴𝑥 ≤ -𝑣) ↔ ∃𝑢 ∈ ℝ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣𝑢)
5430, 53syl6bb 278 . . . . 5 (𝐴 ⊆ ℝ → (∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑥𝑦 ↔ ∃𝑢 ∈ ℝ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣𝑢))
5513, 54anbi12d 618 . . . 4 (𝐴 ⊆ ℝ → ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑥𝑦) ↔ ({𝑤 ∈ ℝ ∣ -𝑤𝐴} ≠ ∅ ∧ ∃𝑢 ∈ ℝ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣𝑢)))
56 ssrab2 3895 . . . . 5 {𝑤 ∈ ℝ ∣ -𝑤𝐴} ⊆ ℝ
57 sup3 11272 . . . . 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 1565 . . . 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 244 . . 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 332 . . . . . . . . 9 (𝐴 ⊆ ℝ → ((𝑦𝐴 → ¬ 𝑦 < 𝑥) ↔ ((𝑦 ∈ ℝ ∧ 𝑦𝐴) → ¬ 𝑦 < 𝑥)))
61 impexp 439 . . . . . . . . 9 (((𝑦 ∈ ℝ ∧ 𝑦𝐴) → ¬ 𝑦 < 𝑥) ↔ (𝑦 ∈ ℝ → (𝑦𝐴 → ¬ 𝑦 < 𝑥)))
6260, 61syl6bb 278 . . . . . . . 8 (𝐴 ⊆ ℝ → ((𝑦𝐴 → ¬ 𝑦 < 𝑥) ↔ (𝑦 ∈ ℝ → (𝑦𝐴 → ¬ 𝑦 < 𝑥))))
6362albidv 2011 . . . . . . 7 (𝐴 ⊆ ℝ → (∀𝑦(𝑦𝐴 → ¬ 𝑦 < 𝑥) ↔ ∀𝑦(𝑦 ∈ ℝ → (𝑦𝐴 → ¬ 𝑦 < 𝑥))))
64 df-ral 3112 . . . . . . 7 (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑦(𝑦𝐴 → ¬ 𝑦 < 𝑥))
65 breq1 4858 . . . . . . . . . . 11 (𝑦 = -𝑣 → (𝑦 < 𝑥 ↔ -𝑣 < 𝑥))
6665notbid 309 . . . . . . . . . 10 (𝑦 = -𝑣 → (¬ 𝑦 < 𝑥 ↔ ¬ -𝑣 < 𝑥))
6723, 66imbi12d 335 . . . . . . . . 9 (𝑦 = -𝑣 → ((𝑦𝐴 → ¬ 𝑦 < 𝑥) ↔ (-𝑣𝐴 → ¬ -𝑣 < 𝑥)))
6821, 22, 67ralxfr 5094 . . . . . . . 8 (∀𝑦 ∈ ℝ (𝑦𝐴 → ¬ 𝑦 < 𝑥) ↔ ∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ -𝑣 < 𝑥))
69 df-ral 3112 . . . . . . . 8 (∀𝑦 ∈ ℝ (𝑦𝐴 → ¬ 𝑦 < 𝑥) ↔ ∀𝑦(𝑦 ∈ ℝ → (𝑦𝐴 → ¬ 𝑦 < 𝑥)))
7068, 69bitr3i 268 . . . . . . 7 (∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ -𝑣 < 𝑥) ↔ ∀𝑦(𝑦 ∈ ℝ → (𝑦𝐴 → ¬ 𝑦 < 𝑥)))
7163, 64, 703bitr4g 305 . . . . . 6 (𝐴 ⊆ ℝ → (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ↔ ∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ -𝑣 < 𝑥)))
72 breq2 4859 . . . . . . . . 9 (𝑦 = -𝑣 → (𝑥 < 𝑦𝑥 < -𝑣))
73 breq2 4859 . . . . . . . . . 10 (𝑦 = -𝑣 → (𝑧 < 𝑦𝑧 < -𝑣))
7473rexbidv 3251 . . . . . . . . 9 (𝑦 = -𝑣 → (∃𝑧𝐴 𝑧 < 𝑦 ↔ ∃𝑧𝐴 𝑧 < -𝑣))
7572, 74imbi12d 335 . . . . . . . 8 (𝑦 = -𝑣 → ((𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ (𝑥 < -𝑣 → ∃𝑧𝐴 𝑧 < -𝑣)))
7621, 22, 75ralxfr 5094 . . . . . . 7 (∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ ∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑧𝐴 𝑧 < -𝑣))
77 ssel 3803 . . . . . . . . . . . . 13 (𝐴 ⊆ ℝ → (𝑧𝐴𝑧 ∈ ℝ))
7877adantrd 481 . . . . . . . . . . . 12 (𝐴 ⊆ ℝ → ((𝑧𝐴𝑧 < -𝑣) → 𝑧 ∈ ℝ))
7978pm4.71rd 554 . . . . . . . . . . 11 (𝐴 ⊆ ℝ → ((𝑧𝐴𝑧 < -𝑣) ↔ (𝑧 ∈ ℝ ∧ (𝑧𝐴𝑧 < -𝑣))))
8079exbidv 2012 . . . . . . . . . 10 (𝐴 ⊆ ℝ → (∃𝑧(𝑧𝐴𝑧 < -𝑣) ↔ ∃𝑧(𝑧 ∈ ℝ ∧ (𝑧𝐴𝑧 < -𝑣))))
81 df-rex 3113 . . . . . . . . . 10 (∃𝑧𝐴 𝑧 < -𝑣 ↔ ∃𝑧(𝑧𝐴𝑧 < -𝑣))
82 renegcl 10636 . . . . . . . . . . . 12 (𝑡 ∈ ℝ → -𝑡 ∈ ℝ)
83 infm3lem 11273 . . . . . . . . . . . 12 (𝑧 ∈ ℝ → ∃𝑡 ∈ ℝ 𝑧 = -𝑡)
84 eleq1 2884 . . . . . . . . . . . . 13 (𝑧 = -𝑡 → (𝑧𝐴 ↔ -𝑡𝐴))
85 breq1 4858 . . . . . . . . . . . . 13 (𝑧 = -𝑡 → (𝑧 < -𝑣 ↔ -𝑡 < -𝑣))
8684, 85anbi12d 618 . . . . . . . . . . . 12 (𝑧 = -𝑡 → ((𝑧𝐴𝑧 < -𝑣) ↔ (-𝑡𝐴 ∧ -𝑡 < -𝑣)))
8782, 83, 86rexxfr 5096 . . . . . . . . . . 11 (∃𝑧 ∈ ℝ (𝑧𝐴𝑧 < -𝑣) ↔ ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣))
88 df-rex 3113 . . . . . . . . . . 11 (∃𝑧 ∈ ℝ (𝑧𝐴𝑧 < -𝑣) ↔ ∃𝑧(𝑧 ∈ ℝ ∧ (𝑧𝐴𝑧 < -𝑣)))
8987, 88bitr3i 268 . . . . . . . . . 10 (∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣) ↔ ∃𝑧(𝑧 ∈ ℝ ∧ (𝑧𝐴𝑧 < -𝑣)))
9080, 81, 893bitr4g 305 . . . . . . . . 9 (𝐴 ⊆ ℝ → (∃𝑧𝐴 𝑧 < -𝑣 ↔ ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣)))
9190imbi2d 331 . . . . . . . 8 (𝐴 ⊆ ℝ → ((𝑥 < -𝑣 → ∃𝑧𝐴 𝑧 < -𝑣) ↔ (𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣))))
9291ralbidv 3185 . . . . . . 7 (𝐴 ⊆ ℝ → (∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑧𝐴 𝑧 < -𝑣) ↔ ∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣))))
9376, 92syl5bb 274 . . . . . 6 (𝐴 ⊆ ℝ → (∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ ∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣))))
9471, 93anbi12d 618 . . . . 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 3251 . . . 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 4859 . . . . . . . . . 10 (𝑥 = -𝑢 → (-𝑣 < 𝑥 ↔ -𝑣 < -𝑢))
9796notbid 309 . . . . . . . . 9 (𝑥 = -𝑢 → (¬ -𝑣 < 𝑥 ↔ ¬ -𝑣 < -𝑢))
9897imbi2d 331 . . . . . . . 8 (𝑥 = -𝑢 → ((-𝑣𝐴 → ¬ -𝑣 < 𝑥) ↔ (-𝑣𝐴 → ¬ -𝑣 < -𝑢)))
9998ralbidv 3185 . . . . . . 7 (𝑥 = -𝑢 → (∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ -𝑣 < 𝑥) ↔ ∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ -𝑣 < -𝑢)))
100 breq1 4858 . . . . . . . . 9 (𝑥 = -𝑢 → (𝑥 < -𝑣 ↔ -𝑢 < -𝑣))
101100imbi1d 332 . . . . . . . 8 (𝑥 = -𝑢 → ((𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣)) ↔ (-𝑢 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣))))
102101ralbidv 3185 . . . . . . 7 (𝑥 = -𝑢 → (∀𝑣 ∈ ℝ (𝑥 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣)) ↔ ∀𝑣 ∈ ℝ (-𝑢 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣))))
10399, 102anbi12d 618 . . . . . 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 5096 . . . . 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 340 . . . . . . . . . . 11 ((𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → ¬ 𝑢 < 𝑣) ↔ ((𝑣 ∈ ℝ ∧ -𝑣𝐴) → ¬ 𝑢 < 𝑣))
106 impexp 439 . . . . . . . . . . 11 (((𝑣 ∈ ℝ ∧ -𝑣𝐴) → ¬ 𝑢 < 𝑣) ↔ (𝑣 ∈ ℝ → (-𝑣𝐴 → ¬ 𝑢 < 𝑣)))
107105, 106bitri 266 . . . . . . . . . 10 ((𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → ¬ 𝑢 < 𝑣) ↔ (𝑣 ∈ ℝ → (-𝑣𝐴 → ¬ 𝑢 < 𝑣)))
108107albii 1904 . . . . . . . . 9 (∀𝑣(𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → ¬ 𝑢 < 𝑣) ↔ ∀𝑣(𝑣 ∈ ℝ → (-𝑣𝐴 → ¬ 𝑢 < 𝑣)))
109 df-ral 3112 . . . . . . . . 9 (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑢 < 𝑣 ↔ ∀𝑣(𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → ¬ 𝑢 < 𝑣))
110 df-ral 3112 . . . . . . . . 9 (∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ 𝑢 < 𝑣) ↔ ∀𝑣(𝑣 ∈ ℝ → (-𝑣𝐴 → ¬ 𝑢 < 𝑣)))
111108, 109, 1103bitr4ri 295 . . . . . . . 8 (∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ 𝑢 < 𝑣) ↔ ∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑢 < 𝑣)
112 ltneg 10820 . . . . . . . . . . 11 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑢 < 𝑣 ↔ -𝑣 < -𝑢))
113112notbid 309 . . . . . . . . . 10 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (¬ 𝑢 < 𝑣 ↔ ¬ -𝑣 < -𝑢))
114113imbi2d 331 . . . . . . . . 9 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((-𝑣𝐴 → ¬ 𝑢 < 𝑣) ↔ (-𝑣𝐴 → ¬ -𝑣 < -𝑢)))
115114ralbidva 3184 . . . . . . . 8 (𝑢 ∈ ℝ → (∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ 𝑢 < 𝑣) ↔ ∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ -𝑣 < -𝑢)))
116111, 115syl5bbr 276 . . . . . . 7 (𝑢 ∈ ℝ → (∀𝑣 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑢 < 𝑣 ↔ ∀𝑣 ∈ ℝ (-𝑣𝐴 → ¬ -𝑣 < -𝑢)))
117 ltneg 10820 . . . . . . . . . 10 ((𝑣 ∈ ℝ ∧ 𝑢 ∈ ℝ) → (𝑣 < 𝑢 ↔ -𝑢 < -𝑣))
118117ancoms 448 . . . . . . . . 9 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (𝑣 < 𝑢 ↔ -𝑢 < -𝑣))
119 negeq 10565 . . . . . . . . . . . . 13 (𝑤 = 𝑡 → -𝑤 = -𝑡)
120119eleq1d 2881 . . . . . . . . . . . 12 (𝑤 = 𝑡 → (-𝑤𝐴 ↔ -𝑡𝐴))
121120rexrab 3577 . . . . . . . . . . 11 (∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣 < 𝑡 ↔ ∃𝑡 ∈ ℝ (-𝑡𝐴𝑣 < 𝑡))
122 ltneg 10820 . . . . . . . . . . . . 13 ((𝑣 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑣 < 𝑡 ↔ -𝑡 < -𝑣))
123122anbi2d 616 . . . . . . . . . . . 12 ((𝑣 ∈ ℝ ∧ 𝑡 ∈ ℝ) → ((-𝑡𝐴𝑣 < 𝑡) ↔ (-𝑡𝐴 ∧ -𝑡 < -𝑣)))
124123rexbidva 3248 . . . . . . . . . . 11 (𝑣 ∈ ℝ → (∃𝑡 ∈ ℝ (-𝑡𝐴𝑣 < 𝑡) ↔ ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣)))
125121, 124syl5bb 274 . . . . . . . . . 10 (𝑣 ∈ ℝ → (∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣 < 𝑡 ↔ ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣)))
126125adantl 469 . . . . . . . . 9 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → (∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣 < 𝑡 ↔ ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣)))
127118, 126imbi12d 335 . . . . . . . 8 ((𝑢 ∈ ℝ ∧ 𝑣 ∈ ℝ) → ((𝑣 < 𝑢 → ∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣 < 𝑡) ↔ (-𝑢 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣))))
128127ralbidva 3184 . . . . . . 7 (𝑢 ∈ ℝ → (∀𝑣 ∈ ℝ (𝑣 < 𝑢 → ∃𝑡 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑣 < 𝑡) ↔ ∀𝑣 ∈ ℝ (-𝑢 < -𝑣 → ∃𝑡 ∈ ℝ (-𝑡𝐴 ∧ -𝑡 < -𝑣))))
129116, 128anbi12d 618 . . . . . 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 3239 . . . . 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 269 . . . 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 278 . . 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 250 . 2 (𝐴 ⊆ ℝ → ((𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦𝐴 𝑥𝑦) → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))))
1341333impib 1137 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 197  wa 384  w3a 1100  wal 1635   = wceq 1637  wex 1859  wcel 2157  wne 2989  wral 3107  wrex 3108  {crab 3111  wss 3780  c0 4127   class class class wbr 4855  cr 10227   < clt 10366  cle 10367  -cneg 10559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4986  ax-nul 4994  ax-pow 5046  ax-pr 5107  ax-un 7186  ax-resscn 10285  ax-1cn 10286  ax-icn 10287  ax-addcl 10288  ax-addrcl 10289  ax-mulcl 10290  ax-mulrcl 10291  ax-mulcom 10292  ax-addass 10293  ax-mulass 10294  ax-distr 10295  ax-i2m1 10296  ax-1ne0 10297  ax-1rid 10298  ax-rnegex 10299  ax-rrecex 10300  ax-cnre 10301  ax-pre-lttri 10302  ax-pre-lttrn 10303  ax-pre-ltadd 10304  ax-pre-mulgt0 10305  ax-pre-sup 10306
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3or 1101  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-nel 3093  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3404  df-sbc 3645  df-csb 3740  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5230  df-po 5243  df-so 5244  df-xp 5328  df-rel 5329  df-cnv 5330  df-co 5331  df-dm 5332  df-rn 5333  df-res 5334  df-ima 5335  df-iota 6071  df-fun 6110  df-fn 6111  df-f 6112  df-f1 6113  df-fo 6114  df-f1o 6115  df-fv 6116  df-riota 6842  df-ov 6884  df-oprab 6885  df-mpt2 6886  df-er 7986  df-en 8200  df-dom 8201  df-sdom 8202  df-pnf 10368  df-mnf 10369  df-xr 10370  df-ltxr 10371  df-le 10372  df-sub 10560  df-neg 10561
This theorem is referenced by:  infrecl  11297  infrenegsup  11298  infregelb  11299  infrelb  11300  xrinfmsslem  12363  gtinf  32643  gtinfOLD  32644  infrglb  40307
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