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Theorem supinfneg 9627
Description: If a set of real numbers has a least upper bound, the set of the negation of those numbers has a greatest lower bound. For a theorem which is similar but only for the boundedness part, see ublbneg 9645. (Contributed by Jim Kingdon, 15-Jan-2022.)
Hypotheses
Ref Expression
supinfneg.ex (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
supinfneg.ss (𝜑𝐴 ⊆ ℝ)
Assertion
Ref Expression
supinfneg (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦)))
Distinct variable groups:   𝑦,𝐴,𝑧,𝑤,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑧,𝑤)

Proof of Theorem supinfneg
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 supinfneg.ex . . . 4 (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
2 breq1 4021 . . . . . . . 8 (𝑎 = 𝑥 → (𝑎 < 𝑦𝑥 < 𝑦))
32notbid 668 . . . . . . 7 (𝑎 = 𝑥 → (¬ 𝑎 < 𝑦 ↔ ¬ 𝑥 < 𝑦))
43ralbidv 2490 . . . . . 6 (𝑎 = 𝑥 → (∀𝑦𝐴 ¬ 𝑎 < 𝑦 ↔ ∀𝑦𝐴 ¬ 𝑥 < 𝑦))
5 breq2 4022 . . . . . . . 8 (𝑎 = 𝑥 → (𝑦 < 𝑎𝑦 < 𝑥))
65imbi1d 231 . . . . . . 7 (𝑎 = 𝑥 → ((𝑦 < 𝑎 → ∃𝑧𝐴 𝑦 < 𝑧) ↔ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
76ralbidv 2490 . . . . . 6 (𝑎 = 𝑥 → (∀𝑦 ∈ ℝ (𝑦 < 𝑎 → ∃𝑧𝐴 𝑦 < 𝑧) ↔ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
84, 7anbi12d 473 . . . . 5 (𝑎 = 𝑥 → ((∀𝑦𝐴 ¬ 𝑎 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑎 → ∃𝑧𝐴 𝑦 < 𝑧)) ↔ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧))))
98cbvrexv 2719 . . . 4 (∃𝑎 ∈ ℝ (∀𝑦𝐴 ¬ 𝑎 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑎 → ∃𝑧𝐴 𝑦 < 𝑧)) ↔ ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
101, 9sylibr 134 . . 3 (𝜑 → ∃𝑎 ∈ ℝ (∀𝑦𝐴 ¬ 𝑎 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑎 → ∃𝑧𝐴 𝑦 < 𝑧)))
11 breq2 4022 . . . . . . 7 (𝑏 = 𝑦 → (𝑎 < 𝑏𝑎 < 𝑦))
1211notbid 668 . . . . . 6 (𝑏 = 𝑦 → (¬ 𝑎 < 𝑏 ↔ ¬ 𝑎 < 𝑦))
1312cbvralv 2718 . . . . 5 (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ↔ ∀𝑦𝐴 ¬ 𝑎 < 𝑦)
14 breq2 4022 . . . . . . . . 9 (𝑐 = 𝑧 → (𝑏 < 𝑐𝑏 < 𝑧))
1514cbvrexv 2719 . . . . . . . 8 (∃𝑐𝐴 𝑏 < 𝑐 ↔ ∃𝑧𝐴 𝑏 < 𝑧)
1615imbi2i 226 . . . . . . 7 ((𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐) ↔ (𝑏 < 𝑎 → ∃𝑧𝐴 𝑏 < 𝑧))
1716ralbii 2496 . . . . . 6 (∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐) ↔ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑧𝐴 𝑏 < 𝑧))
18 breq1 4021 . . . . . . . 8 (𝑏 = 𝑦 → (𝑏 < 𝑎𝑦 < 𝑎))
19 breq1 4021 . . . . . . . . 9 (𝑏 = 𝑦 → (𝑏 < 𝑧𝑦 < 𝑧))
2019rexbidv 2491 . . . . . . . 8 (𝑏 = 𝑦 → (∃𝑧𝐴 𝑏 < 𝑧 ↔ ∃𝑧𝐴 𝑦 < 𝑧))
2118, 20imbi12d 234 . . . . . . 7 (𝑏 = 𝑦 → ((𝑏 < 𝑎 → ∃𝑧𝐴 𝑏 < 𝑧) ↔ (𝑦 < 𝑎 → ∃𝑧𝐴 𝑦 < 𝑧)))
2221cbvralv 2718 . . . . . 6 (∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑧𝐴 𝑏 < 𝑧) ↔ ∀𝑦 ∈ ℝ (𝑦 < 𝑎 → ∃𝑧𝐴 𝑦 < 𝑧))
2317, 22bitri 184 . . . . 5 (∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐) ↔ ∀𝑦 ∈ ℝ (𝑦 < 𝑎 → ∃𝑧𝐴 𝑦 < 𝑧))
2413, 23anbi12i 460 . . . 4 ((∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ↔ (∀𝑦𝐴 ¬ 𝑎 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑎 → ∃𝑧𝐴 𝑦 < 𝑧)))
2524rexbii 2497 . . 3 (∃𝑎 ∈ ℝ (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ↔ ∃𝑎 ∈ ℝ (∀𝑦𝐴 ¬ 𝑎 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑎 → ∃𝑧𝐴 𝑦 < 𝑧)))
2610, 25sylibr 134 . 2 (𝜑 → ∃𝑎 ∈ ℝ (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)))
27 renegcl 8249 . . . . . 6 (𝑎 ∈ ℝ → -𝑎 ∈ ℝ)
2827ad2antlr 489 . . . . 5 (((𝜑𝑎 ∈ ℝ) ∧ (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐))) → -𝑎 ∈ ℝ)
29 simplr 528 . . . . . 6 (((𝜑𝑎 ∈ ℝ) ∧ (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐))) → 𝑎 ∈ ℝ)
30 simprl 529 . . . . . 6 (((𝜑𝑎 ∈ ℝ) ∧ (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐))) → ∀𝑏𝐴 ¬ 𝑎 < 𝑏)
31 elrabi 2905 . . . . . . . . . . . 12 (𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → 𝑦 ∈ ℝ)
32 negeq 8181 . . . . . . . . . . . . . . 15 (𝑤 = 𝑦 → -𝑤 = -𝑦)
3332eleq1d 2258 . . . . . . . . . . . . . 14 (𝑤 = 𝑦 → (-𝑤𝐴 ↔ -𝑦𝐴))
3433elrab3 2909 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ → (𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ↔ -𝑦𝐴))
3534biimpd 144 . . . . . . . . . . . 12 (𝑦 ∈ ℝ → (𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → -𝑦𝐴))
3631, 35mpcom 36 . . . . . . . . . . 11 (𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → -𝑦𝐴)
37 breq2 4022 . . . . . . . . . . . . 13 (𝑏 = -𝑦 → (𝑎 < 𝑏𝑎 < -𝑦))
3837notbid 668 . . . . . . . . . . . 12 (𝑏 = -𝑦 → (¬ 𝑎 < 𝑏 ↔ ¬ 𝑎 < -𝑦))
3938rspcv 2852 . . . . . . . . . . 11 (-𝑦𝐴 → (∀𝑏𝐴 ¬ 𝑎 < 𝑏 → ¬ 𝑎 < -𝑦))
4036, 39syl 14 . . . . . . . . . 10 (𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → (∀𝑏𝐴 ¬ 𝑎 < 𝑏 → ¬ 𝑎 < -𝑦))
4140adantr 276 . . . . . . . . 9 ((𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ∧ 𝑎 ∈ ℝ) → (∀𝑏𝐴 ¬ 𝑎 < 𝑏 → ¬ 𝑎 < -𝑦))
42 ltnegcon2 8452 . . . . . . . . . . 11 ((𝑦 ∈ ℝ ∧ 𝑎 ∈ ℝ) → (𝑦 < -𝑎𝑎 < -𝑦))
4342notbid 668 . . . . . . . . . 10 ((𝑦 ∈ ℝ ∧ 𝑎 ∈ ℝ) → (¬ 𝑦 < -𝑎 ↔ ¬ 𝑎 < -𝑦))
4431, 43sylan 283 . . . . . . . . 9 ((𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ∧ 𝑎 ∈ ℝ) → (¬ 𝑦 < -𝑎 ↔ ¬ 𝑎 < -𝑦))
4541, 44sylibrd 169 . . . . . . . 8 ((𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ∧ 𝑎 ∈ ℝ) → (∀𝑏𝐴 ¬ 𝑎 < 𝑏 → ¬ 𝑦 < -𝑎))
4645ancoms 268 . . . . . . 7 ((𝑎 ∈ ℝ ∧ 𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}) → (∀𝑏𝐴 ¬ 𝑎 < 𝑏 → ¬ 𝑦 < -𝑎))
4746ralrimdva 2570 . . . . . 6 (𝑎 ∈ ℝ → (∀𝑏𝐴 ¬ 𝑎 < 𝑏 → ∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑦 < -𝑎))
4829, 30, 47sylc 62 . . . . 5 (((𝜑𝑎 ∈ ℝ) ∧ (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐))) → ∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑦 < -𝑎)
49 nfv 1539 . . . . . . . . . . . 12 𝑐(𝜑𝑎 ∈ ℝ)
50 nfcv 2332 . . . . . . . . . . . . 13 𝑐
51 nfv 1539 . . . . . . . . . . . . . 14 𝑐 𝑏 < 𝑎
52 nfre1 2533 . . . . . . . . . . . . . 14 𝑐𝑐𝐴 𝑏 < 𝑐
5351, 52nfim 1583 . . . . . . . . . . . . 13 𝑐(𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)
5450, 53nfralya 2530 . . . . . . . . . . . 12 𝑐𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)
5549, 54nfan 1576 . . . . . . . . . . 11 𝑐((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐))
56 nfv 1539 . . . . . . . . . . 11 𝑐 𝑦 ∈ ℝ
5755, 56nfan 1576 . . . . . . . . . 10 𝑐(((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ)
58 nfv 1539 . . . . . . . . . 10 𝑐-𝑎 < 𝑦
5957, 58nfan 1576 . . . . . . . . 9 𝑐((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦)
60 simplr 528 . . . . . . . . . . . . 13 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐𝐴) ∧ -𝑦 < 𝑐) → 𝑐𝐴)
61 supinfneg.ss . . . . . . . . . . . . . . 15 (𝜑𝐴 ⊆ ℝ)
6261sseld 3169 . . . . . . . . . . . . . 14 (𝜑 → (𝑐𝐴𝑐 ∈ ℝ))
6362ad6antr 498 . . . . . . . . . . . . 13 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐𝐴) ∧ -𝑦 < 𝑐) → (𝑐𝐴𝑐 ∈ ℝ))
6460, 63mpd 13 . . . . . . . . . . . 12 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐𝐴) ∧ -𝑦 < 𝑐) → 𝑐 ∈ ℝ)
6564renegcld 8368 . . . . . . . . . . 11 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐𝐴) ∧ -𝑦 < 𝑐) → -𝑐 ∈ ℝ)
6664recnd 8017 . . . . . . . . . . . . 13 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐𝐴) ∧ -𝑦 < 𝑐) → 𝑐 ∈ ℂ)
6766negnegd 8290 . . . . . . . . . . . 12 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐𝐴) ∧ -𝑦 < 𝑐) → --𝑐 = 𝑐)
6867, 60eqeltrd 2266 . . . . . . . . . . 11 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐𝐴) ∧ -𝑦 < 𝑐) → --𝑐𝐴)
69 negeq 8181 . . . . . . . . . . . . 13 (𝑤 = -𝑐 → -𝑤 = --𝑐)
7069eleq1d 2258 . . . . . . . . . . . 12 (𝑤 = -𝑐 → (-𝑤𝐴 ↔ --𝑐𝐴))
7170elrab 2908 . . . . . . . . . . 11 (-𝑐 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ↔ (-𝑐 ∈ ℝ ∧ --𝑐𝐴))
7265, 68, 71sylanbrc 417 . . . . . . . . . 10 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐𝐴) ∧ -𝑦 < 𝑐) → -𝑐 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴})
73 simp-4r 542 . . . . . . . . . . 11 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐𝐴) ∧ -𝑦 < 𝑐) → 𝑦 ∈ ℝ)
74 simpr 110 . . . . . . . . . . 11 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐𝐴) ∧ -𝑦 < 𝑐) → -𝑦 < 𝑐)
7573, 64, 74ltnegcon1d 8513 . . . . . . . . . 10 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐𝐴) ∧ -𝑦 < 𝑐) → -𝑐 < 𝑦)
76 breq1 4021 . . . . . . . . . . 11 (𝑧 = -𝑐 → (𝑧 < 𝑦 ↔ -𝑐 < 𝑦))
7776rspcev 2856 . . . . . . . . . 10 ((-𝑐 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ∧ -𝑐 < 𝑦) → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦)
7872, 75, 77syl2anc 411 . . . . . . . . 9 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐𝐴) ∧ -𝑦 < 𝑐) → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦)
79 simpllr 534 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) → 𝑎 ∈ ℝ)
80 simpr 110 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
81 simplr 528 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) → ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐))
8279, 80, 81jca31 309 . . . . . . . . . 10 ((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) → ((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)))
83 ltnegcon1 8451 . . . . . . . . . . . . 13 ((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (-𝑎 < 𝑦 ↔ -𝑦 < 𝑎))
8483adantr 276 . . . . . . . . . . . 12 (((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) → (-𝑎 < 𝑦 ↔ -𝑦 < 𝑎))
85 renegcl 8249 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
86 breq1 4021 . . . . . . . . . . . . . . . . 17 (𝑏 = -𝑦 → (𝑏 < 𝑎 ↔ -𝑦 < 𝑎))
87 breq1 4021 . . . . . . . . . . . . . . . . . 18 (𝑏 = -𝑦 → (𝑏 < 𝑐 ↔ -𝑦 < 𝑐))
8887rexbidv 2491 . . . . . . . . . . . . . . . . 17 (𝑏 = -𝑦 → (∃𝑐𝐴 𝑏 < 𝑐 ↔ ∃𝑐𝐴 -𝑦 < 𝑐))
8986, 88imbi12d 234 . . . . . . . . . . . . . . . 16 (𝑏 = -𝑦 → ((𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐) ↔ (-𝑦 < 𝑎 → ∃𝑐𝐴 -𝑦 < 𝑐)))
9089rspcv 2852 . . . . . . . . . . . . . . 15 (-𝑦 ∈ ℝ → (∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐) → (-𝑦 < 𝑎 → ∃𝑐𝐴 -𝑦 < 𝑐)))
9185, 90syl 14 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ → (∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐) → (-𝑦 < 𝑎 → ∃𝑐𝐴 -𝑦 < 𝑐)))
9291adantl 277 . . . . . . . . . . . . 13 ((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐) → (-𝑦 < 𝑎 → ∃𝑐𝐴 -𝑦 < 𝑐)))
9392imp 124 . . . . . . . . . . . 12 (((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) → (-𝑦 < 𝑎 → ∃𝑐𝐴 -𝑦 < 𝑐))
9484, 93sylbid 150 . . . . . . . . . . 11 (((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) → (-𝑎 < 𝑦 → ∃𝑐𝐴 -𝑦 < 𝑐))
9594imp 124 . . . . . . . . . 10 ((((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ -𝑎 < 𝑦) → ∃𝑐𝐴 -𝑦 < 𝑐)
9682, 95sylan 283 . . . . . . . . 9 (((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) → ∃𝑐𝐴 -𝑦 < 𝑐)
9759, 78, 96r19.29af 2631 . . . . . . . 8 (((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦)
9897ex 115 . . . . . . 7 ((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) → (-𝑎 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦))
9998ralrimiva 2563 . . . . . 6 (((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) → ∀𝑦 ∈ ℝ (-𝑎 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦))
10099adantrl 478 . . . . 5 (((𝜑𝑎 ∈ ℝ) ∧ (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐))) → ∀𝑦 ∈ ℝ (-𝑎 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦))
101 breq2 4022 . . . . . . . . 9 (𝑥 = -𝑎 → (𝑦 < 𝑥𝑦 < -𝑎))
102101notbid 668 . . . . . . . 8 (𝑥 = -𝑎 → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < -𝑎))
103102ralbidv 2490 . . . . . . 7 (𝑥 = -𝑎 → (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑦 < 𝑥 ↔ ∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑦 < -𝑎))
104 breq1 4021 . . . . . . . . 9 (𝑥 = -𝑎 → (𝑥 < 𝑦 ↔ -𝑎 < 𝑦))
105104imbi1d 231 . . . . . . . 8 (𝑥 = -𝑎 → ((𝑥 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦) ↔ (-𝑎 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦)))
106105ralbidv 2490 . . . . . . 7 (𝑥 = -𝑎 → (∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦) ↔ ∀𝑦 ∈ ℝ (-𝑎 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦)))
107103, 106anbi12d 473 . . . . . 6 (𝑥 = -𝑎 → ((∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦)) ↔ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑦 < -𝑎 ∧ ∀𝑦 ∈ ℝ (-𝑎 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦))))
108107rspcev 2856 . . . . 5 ((-𝑎 ∈ ℝ ∧ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑦 < -𝑎 ∧ ∀𝑦 ∈ ℝ (-𝑎 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦)))
10928, 48, 100, 108syl12anc 1247 . . . 4 (((𝜑𝑎 ∈ ℝ) ∧ (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦)))
110109ex 115 . . 3 ((𝜑𝑎 ∈ ℝ) → ((∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦))))
111110rexlimdva 2607 . 2 (𝜑 → (∃𝑎 ∈ ℝ (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦))))
11226, 111mpd 13 1 (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105   = wceq 1364  wcel 2160  wral 2468  wrex 2469  {crab 2472  wss 3144   class class class wbr 4018  cr 7841   < clt 8023  -cneg 8160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7933  ax-resscn 7934  ax-1cn 7935  ax-1re 7936  ax-icn 7937  ax-addcl 7938  ax-addrcl 7939  ax-mulcl 7940  ax-addcom 7942  ax-addass 7944  ax-distr 7946  ax-i2m1 7947  ax-0id 7950  ax-rnegex 7951  ax-cnre 7953  ax-pre-ltadd 7958
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-iota 5196  df-fun 5237  df-fv 5243  df-riota 5852  df-ov 5900  df-oprab 5901  df-mpo 5902  df-pnf 8025  df-mnf 8026  df-ltxr 8028  df-sub 8161  df-neg 8162
This theorem is referenced by:  supminfex  9629  infssuzex  11985
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