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Theorem supinfneg 9541
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 9559. (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 3990 . . . . . . . 8 (𝑎 = 𝑥 → (𝑎 < 𝑦𝑥 < 𝑦))
32notbid 662 . . . . . . 7 (𝑎 = 𝑥 → (¬ 𝑎 < 𝑦 ↔ ¬ 𝑥 < 𝑦))
43ralbidv 2470 . . . . . 6 (𝑎 = 𝑥 → (∀𝑦𝐴 ¬ 𝑎 < 𝑦 ↔ ∀𝑦𝐴 ¬ 𝑥 < 𝑦))
5 breq2 3991 . . . . . . . 8 (𝑎 = 𝑥 → (𝑦 < 𝑎𝑦 < 𝑥))
65imbi1d 230 . . . . . . 7 (𝑎 = 𝑥 → ((𝑦 < 𝑎 → ∃𝑧𝐴 𝑦 < 𝑧) ↔ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
76ralbidv 2470 . . . . . 6 (𝑎 = 𝑥 → (∀𝑦 ∈ ℝ (𝑦 < 𝑎 → ∃𝑧𝐴 𝑦 < 𝑧) ↔ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧)))
84, 7anbi12d 470 . . . . 5 (𝑎 = 𝑥 → ((∀𝑦𝐴 ¬ 𝑎 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑎 → ∃𝑧𝐴 𝑦 < 𝑧)) ↔ (∀𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧𝐴 𝑦 < 𝑧))))
98cbvrexv 2697 . . . 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 133 . . 3 (𝜑 → ∃𝑎 ∈ ℝ (∀𝑦𝐴 ¬ 𝑎 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑎 → ∃𝑧𝐴 𝑦 < 𝑧)))
11 breq2 3991 . . . . . . 7 (𝑏 = 𝑦 → (𝑎 < 𝑏𝑎 < 𝑦))
1211notbid 662 . . . . . 6 (𝑏 = 𝑦 → (¬ 𝑎 < 𝑏 ↔ ¬ 𝑎 < 𝑦))
1312cbvralv 2696 . . . . 5 (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ↔ ∀𝑦𝐴 ¬ 𝑎 < 𝑦)
14 breq2 3991 . . . . . . . . 9 (𝑐 = 𝑧 → (𝑏 < 𝑐𝑏 < 𝑧))
1514cbvrexv 2697 . . . . . . . 8 (∃𝑐𝐴 𝑏 < 𝑐 ↔ ∃𝑧𝐴 𝑏 < 𝑧)
1615imbi2i 225 . . . . . . 7 ((𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐) ↔ (𝑏 < 𝑎 → ∃𝑧𝐴 𝑏 < 𝑧))
1716ralbii 2476 . . . . . 6 (∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐) ↔ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑧𝐴 𝑏 < 𝑧))
18 breq1 3990 . . . . . . . 8 (𝑏 = 𝑦 → (𝑏 < 𝑎𝑦 < 𝑎))
19 breq1 3990 . . . . . . . . 9 (𝑏 = 𝑦 → (𝑏 < 𝑧𝑦 < 𝑧))
2019rexbidv 2471 . . . . . . . 8 (𝑏 = 𝑦 → (∃𝑧𝐴 𝑏 < 𝑧 ↔ ∃𝑧𝐴 𝑦 < 𝑧))
2118, 20imbi12d 233 . . . . . . 7 (𝑏 = 𝑦 → ((𝑏 < 𝑎 → ∃𝑧𝐴 𝑏 < 𝑧) ↔ (𝑦 < 𝑎 → ∃𝑧𝐴 𝑦 < 𝑧)))
2221cbvralv 2696 . . . . . 6 (∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑧𝐴 𝑏 < 𝑧) ↔ ∀𝑦 ∈ ℝ (𝑦 < 𝑎 → ∃𝑧𝐴 𝑦 < 𝑧))
2317, 22bitri 183 . . . . 5 (∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐) ↔ ∀𝑦 ∈ ℝ (𝑦 < 𝑎 → ∃𝑧𝐴 𝑦 < 𝑧))
2413, 23anbi12i 457 . . . 4 ((∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ↔ (∀𝑦𝐴 ¬ 𝑎 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑎 → ∃𝑧𝐴 𝑦 < 𝑧)))
2524rexbii 2477 . . 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 133 . 2 (𝜑 → ∃𝑎 ∈ ℝ (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)))
27 renegcl 8167 . . . . . 6 (𝑎 ∈ ℝ → -𝑎 ∈ ℝ)
2827ad2antlr 486 . . . . 5 (((𝜑𝑎 ∈ ℝ) ∧ (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐))) → -𝑎 ∈ ℝ)
29 simplr 525 . . . . . 6 (((𝜑𝑎 ∈ ℝ) ∧ (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐))) → 𝑎 ∈ ℝ)
30 simprl 526 . . . . . 6 (((𝜑𝑎 ∈ ℝ) ∧ (∀𝑏𝐴 ¬ 𝑎 < 𝑏 ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐))) → ∀𝑏𝐴 ¬ 𝑎 < 𝑏)
31 elrabi 2883 . . . . . . . . . . . 12 (𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → 𝑦 ∈ ℝ)
32 negeq 8099 . . . . . . . . . . . . . . 15 (𝑤 = 𝑦 → -𝑤 = -𝑦)
3332eleq1d 2239 . . . . . . . . . . . . . 14 (𝑤 = 𝑦 → (-𝑤𝐴 ↔ -𝑦𝐴))
3433elrab3 2887 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ → (𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ↔ -𝑦𝐴))
3534biimpd 143 . . . . . . . . . . . 12 (𝑦 ∈ ℝ → (𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → -𝑦𝐴))
3631, 35mpcom 36 . . . . . . . . . . 11 (𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → -𝑦𝐴)
37 breq2 3991 . . . . . . . . . . . . 13 (𝑏 = -𝑦 → (𝑎 < 𝑏𝑎 < -𝑦))
3837notbid 662 . . . . . . . . . . . 12 (𝑏 = -𝑦 → (¬ 𝑎 < 𝑏 ↔ ¬ 𝑎 < -𝑦))
3938rspcv 2830 . . . . . . . . . . 11 (-𝑦𝐴 → (∀𝑏𝐴 ¬ 𝑎 < 𝑏 → ¬ 𝑎 < -𝑦))
4036, 39syl 14 . . . . . . . . . 10 (𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → (∀𝑏𝐴 ¬ 𝑎 < 𝑏 → ¬ 𝑎 < -𝑦))
4140adantr 274 . . . . . . . . 9 ((𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ∧ 𝑎 ∈ ℝ) → (∀𝑏𝐴 ¬ 𝑎 < 𝑏 → ¬ 𝑎 < -𝑦))
42 ltnegcon2 8370 . . . . . . . . . . 11 ((𝑦 ∈ ℝ ∧ 𝑎 ∈ ℝ) → (𝑦 < -𝑎𝑎 < -𝑦))
4342notbid 662 . . . . . . . . . 10 ((𝑦 ∈ ℝ ∧ 𝑎 ∈ ℝ) → (¬ 𝑦 < -𝑎 ↔ ¬ 𝑎 < -𝑦))
4431, 43sylan 281 . . . . . . . . 9 ((𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ∧ 𝑎 ∈ ℝ) → (¬ 𝑦 < -𝑎 ↔ ¬ 𝑎 < -𝑦))
4541, 44sylibrd 168 . . . . . . . 8 ((𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ∧ 𝑎 ∈ ℝ) → (∀𝑏𝐴 ¬ 𝑎 < 𝑏 → ¬ 𝑦 < -𝑎))
4645ancoms 266 . . . . . . 7 ((𝑎 ∈ ℝ ∧ 𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}) → (∀𝑏𝐴 ¬ 𝑎 < 𝑏 → ¬ 𝑦 < -𝑎))
4746ralrimdva 2550 . . . . . 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 1521 . . . . . . . . . . . 12 𝑐(𝜑𝑎 ∈ ℝ)
50 nfcv 2312 . . . . . . . . . . . . 13 𝑐
51 nfv 1521 . . . . . . . . . . . . . 14 𝑐 𝑏 < 𝑎
52 nfre1 2513 . . . . . . . . . . . . . 14 𝑐𝑐𝐴 𝑏 < 𝑐
5351, 52nfim 1565 . . . . . . . . . . . . 13 𝑐(𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)
5450, 53nfralya 2510 . . . . . . . . . . . 12 𝑐𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)
5549, 54nfan 1558 . . . . . . . . . . 11 𝑐((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐))
56 nfv 1521 . . . . . . . . . . 11 𝑐 𝑦 ∈ ℝ
5755, 56nfan 1558 . . . . . . . . . 10 𝑐(((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ)
58 nfv 1521 . . . . . . . . . 10 𝑐-𝑎 < 𝑦
5957, 58nfan 1558 . . . . . . . . 9 𝑐((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦)
60 simplr 525 . . . . . . . . . . . . 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 3146 . . . . . . . . . . . . . 14 (𝜑 → (𝑐𝐴𝑐 ∈ ℝ))
6362ad6antr 495 . . . . . . . . . . . . 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 8286 . . . . . . . . . . 11 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐𝐴) ∧ -𝑦 < 𝑐) → -𝑐 ∈ ℝ)
6664recnd 7935 . . . . . . . . . . . . 13 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐𝐴) ∧ -𝑦 < 𝑐) → 𝑐 ∈ ℂ)
6766negnegd 8208 . . . . . . . . . . . 12 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐𝐴) ∧ -𝑦 < 𝑐) → --𝑐 = 𝑐)
6867, 60eqeltrd 2247 . . . . . . . . . . 11 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐𝐴) ∧ -𝑦 < 𝑐) → --𝑐𝐴)
69 negeq 8099 . . . . . . . . . . . . 13 (𝑤 = -𝑐 → -𝑤 = --𝑐)
7069eleq1d 2239 . . . . . . . . . . . 12 (𝑤 = -𝑐 → (-𝑤𝐴 ↔ --𝑐𝐴))
7170elrab 2886 . . . . . . . . . . 11 (-𝑐 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ↔ (-𝑐 ∈ ℝ ∧ --𝑐𝐴))
7265, 68, 71sylanbrc 415 . . . . . . . . . 10 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐𝐴) ∧ -𝑦 < 𝑐) → -𝑐 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴})
73 simp-4r 537 . . . . . . . . . . 11 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐𝐴) ∧ -𝑦 < 𝑐) → 𝑦 ∈ ℝ)
74 simpr 109 . . . . . . . . . . 11 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐𝐴) ∧ -𝑦 < 𝑐) → -𝑦 < 𝑐)
7573, 64, 74ltnegcon1d 8431 . . . . . . . . . 10 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) ∧ 𝑐𝐴) ∧ -𝑦 < 𝑐) → -𝑐 < 𝑦)
76 breq1 3990 . . . . . . . . . . 11 (𝑧 = -𝑐 → (𝑧 < 𝑦 ↔ -𝑐 < 𝑦))
7776rspcev 2834 . . . . . . . . . 10 ((-𝑐 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ∧ -𝑐 < 𝑦) → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦)
7872, 75, 77syl2anc 409 . . . . . . . . 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 529 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) → 𝑎 ∈ ℝ)
80 simpr 109 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
81 simplr 525 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) → ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐))
8279, 80, 81jca31 307 . . . . . . . . . 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 8369 . . . . . . . . . . . . 13 ((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (-𝑎 < 𝑦 ↔ -𝑦 < 𝑎))
8483adantr 274 . . . . . . . . . . . 12 (((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) → (-𝑎 < 𝑦 ↔ -𝑦 < 𝑎))
85 renegcl 8167 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
86 breq1 3990 . . . . . . . . . . . . . . . . 17 (𝑏 = -𝑦 → (𝑏 < 𝑎 ↔ -𝑦 < 𝑎))
87 breq1 3990 . . . . . . . . . . . . . . . . . 18 (𝑏 = -𝑦 → (𝑏 < 𝑐 ↔ -𝑦 < 𝑐))
8887rexbidv 2471 . . . . . . . . . . . . . . . . 17 (𝑏 = -𝑦 → (∃𝑐𝐴 𝑏 < 𝑐 ↔ ∃𝑐𝐴 -𝑦 < 𝑐))
8986, 88imbi12d 233 . . . . . . . . . . . . . . . 16 (𝑏 = -𝑦 → ((𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐) ↔ (-𝑦 < 𝑎 → ∃𝑐𝐴 -𝑦 < 𝑐)))
9089rspcv 2830 . . . . . . . . . . . . . . 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 275 . . . . . . . . . . . . 13 ((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐) → (-𝑦 < 𝑎 → ∃𝑐𝐴 -𝑦 < 𝑐)))
9392imp 123 . . . . . . . . . . . 12 (((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) → (-𝑦 < 𝑎 → ∃𝑐𝐴 -𝑦 < 𝑐))
9484, 93sylbid 149 . . . . . . . . . . 11 (((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) → (-𝑎 < 𝑦 → ∃𝑐𝐴 -𝑦 < 𝑐))
9594imp 123 . . . . . . . . . 10 ((((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ -𝑎 < 𝑦) → ∃𝑐𝐴 -𝑦 < 𝑐)
9682, 95sylan 281 . . . . . . . . 9 (((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) → ∃𝑐𝐴 -𝑦 < 𝑐)
9759, 78, 96r19.29af 2611 . . . . . . . 8 (((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) ∧ -𝑎 < 𝑦) → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦)
9897ex 114 . . . . . . 7 ((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) ∧ 𝑦 ∈ ℝ) → (-𝑎 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦))
9998ralrimiva 2543 . . . . . 6 (((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑏 < 𝑎 → ∃𝑐𝐴 𝑏 < 𝑐)) → ∀𝑦 ∈ ℝ (-𝑎 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦))
10099adantrl 475 . . . . 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 3991 . . . . . . . . 9 (𝑥 = -𝑎 → (𝑦 < 𝑥𝑦 < -𝑎))
102101notbid 662 . . . . . . . 8 (𝑥 = -𝑎 → (¬ 𝑦 < 𝑥 ↔ ¬ 𝑦 < -𝑎))
103102ralbidv 2470 . . . . . . 7 (𝑥 = -𝑎 → (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑦 < 𝑥 ↔ ∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑦 < -𝑎))
104 breq1 3990 . . . . . . . . 9 (𝑥 = -𝑎 → (𝑥 < 𝑦 ↔ -𝑎 < 𝑦))
105104imbi1d 230 . . . . . . . 8 (𝑥 = -𝑎 → ((𝑥 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦) ↔ (-𝑎 < 𝑦 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑧 < 𝑦)))
106105ralbidv 2470 . . . . . . 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 470 . . . . . 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 2834 . . . . 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 1231 . . . 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 114 . . 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 2587 . 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 103  wb 104   = wceq 1348  wcel 2141  wral 2448  wrex 2449  {crab 2452  wss 3121   class class class wbr 3987  cr 7760   < clt 7941  -cneg 8078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-setind 4519  ax-cnex 7852  ax-resscn 7853  ax-1cn 7854  ax-1re 7855  ax-icn 7856  ax-addcl 7857  ax-addrcl 7858  ax-mulcl 7859  ax-addcom 7861  ax-addass 7863  ax-distr 7865  ax-i2m1 7866  ax-0id 7869  ax-rnegex 7870  ax-cnre 7872  ax-pre-ltadd 7877
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-br 3988  df-opab 4049  df-id 4276  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-iota 5158  df-fun 5198  df-fv 5204  df-riota 5806  df-ov 5853  df-oprab 5854  df-mpo 5855  df-pnf 7943  df-mnf 7944  df-ltxr 7946  df-sub 8079  df-neg 8080
This theorem is referenced by:  supminfex  9543  infssuzex  11891
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