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Theorem infsupneg 9009
Description: If a set of real numbers has a greatest lower bound, the set of the negation of those numbers has a least upper bound. To go in the other direction see supinfneg 9008. (Contributed by Jim Kingdon, 15-Jan-2022.)
Hypotheses
Ref Expression
infsupneg.ex (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
infsupneg.ss (𝜑𝐴 ⊆ ℝ)
Assertion
Ref Expression
infsupneg (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑦 < 𝑧)))
Distinct variable groups:   𝑦,𝐴,𝑧,𝑤,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑧,𝑤)

Proof of Theorem infsupneg
Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 infsupneg.ex . . . 4 (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
2 breq2 3824 . . . . . . . 8 (𝑎 = 𝑥 → (𝑦 < 𝑎𝑦 < 𝑥))
32notbid 625 . . . . . . 7 (𝑎 = 𝑥 → (¬ 𝑦 < 𝑎 ↔ ¬ 𝑦 < 𝑥))
43ralbidv 2376 . . . . . 6 (𝑎 = 𝑥 → (∀𝑦𝐴 ¬ 𝑦 < 𝑎 ↔ ∀𝑦𝐴 ¬ 𝑦 < 𝑥))
5 breq1 3823 . . . . . . . 8 (𝑎 = 𝑥 → (𝑎 < 𝑦𝑥 < 𝑦))
65imbi1d 229 . . . . . . 7 (𝑎 = 𝑥 → ((𝑎 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
76ralbidv 2376 . . . . . 6 (𝑎 = 𝑥 → (∀𝑦 ∈ ℝ (𝑎 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦) ↔ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
84, 7anbi12d 457 . . . . 5 (𝑎 = 𝑥 → ((∀𝑦𝐴 ¬ 𝑦 < 𝑎 ∧ ∀𝑦 ∈ ℝ (𝑎 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) ↔ (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))))
98cbvrexv 2587 . . . 4 (∃𝑎 ∈ ℝ (∀𝑦𝐴 ¬ 𝑦 < 𝑎 ∧ ∀𝑦 ∈ ℝ (𝑎 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)) ↔ ∃𝑥 ∈ ℝ (∀𝑦𝐴 ¬ 𝑦 < 𝑥 ∧ ∀𝑦 ∈ ℝ (𝑥 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
101, 9sylibr 132 . . 3 (𝜑 → ∃𝑎 ∈ ℝ (∀𝑦𝐴 ¬ 𝑦 < 𝑎 ∧ ∀𝑦 ∈ ℝ (𝑎 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
11 breq1 3823 . . . . . . 7 (𝑏 = 𝑦 → (𝑏 < 𝑎𝑦 < 𝑎))
1211notbid 625 . . . . . 6 (𝑏 = 𝑦 → (¬ 𝑏 < 𝑎 ↔ ¬ 𝑦 < 𝑎))
1312cbvralv 2586 . . . . 5 (∀𝑏𝐴 ¬ 𝑏 < 𝑎 ↔ ∀𝑦𝐴 ¬ 𝑦 < 𝑎)
14 breq1 3823 . . . . . . . . 9 (𝑐 = 𝑧 → (𝑐 < 𝑏𝑧 < 𝑏))
1514cbvrexv 2587 . . . . . . . 8 (∃𝑐𝐴 𝑐 < 𝑏 ↔ ∃𝑧𝐴 𝑧 < 𝑏)
1615imbi2i 224 . . . . . . 7 ((𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏) ↔ (𝑎 < 𝑏 → ∃𝑧𝐴 𝑧 < 𝑏))
1716ralbii 2380 . . . . . 6 (∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏) ↔ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑧𝐴 𝑧 < 𝑏))
18 breq2 3824 . . . . . . . 8 (𝑏 = 𝑦 → (𝑎 < 𝑏𝑎 < 𝑦))
19 breq2 3824 . . . . . . . . 9 (𝑏 = 𝑦 → (𝑧 < 𝑏𝑧 < 𝑦))
2019rexbidv 2377 . . . . . . . 8 (𝑏 = 𝑦 → (∃𝑧𝐴 𝑧 < 𝑏 ↔ ∃𝑧𝐴 𝑧 < 𝑦))
2118, 20imbi12d 232 . . . . . . 7 (𝑏 = 𝑦 → ((𝑎 < 𝑏 → ∃𝑧𝐴 𝑧 < 𝑏) ↔ (𝑎 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
2221cbvralv 2586 . . . . . 6 (∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑧𝐴 𝑧 < 𝑏) ↔ ∀𝑦 ∈ ℝ (𝑎 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))
2317, 22bitri 182 . . . . 5 (∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏) ↔ ∀𝑦 ∈ ℝ (𝑎 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦))
2413, 23anbi12i 448 . . . 4 ((∀𝑏𝐴 ¬ 𝑏 < 𝑎 ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) ↔ (∀𝑦𝐴 ¬ 𝑦 < 𝑎 ∧ ∀𝑦 ∈ ℝ (𝑎 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
2524rexbii 2381 . . 3 (∃𝑎 ∈ ℝ (∀𝑏𝐴 ¬ 𝑏 < 𝑎 ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) ↔ ∃𝑎 ∈ ℝ (∀𝑦𝐴 ¬ 𝑦 < 𝑎 ∧ ∀𝑦 ∈ ℝ (𝑎 < 𝑦 → ∃𝑧𝐴 𝑧 < 𝑦)))
2610, 25sylibr 132 . 2 (𝜑 → ∃𝑎 ∈ ℝ (∀𝑏𝐴 ¬ 𝑏 < 𝑎 ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)))
27 renegcl 7680 . . . . . 6 (𝑎 ∈ ℝ → -𝑎 ∈ ℝ)
2827ad2antlr 473 . . . . 5 (((𝜑𝑎 ∈ ℝ) ∧ (∀𝑏𝐴 ¬ 𝑏 < 𝑎 ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏))) → -𝑎 ∈ ℝ)
29 simplr 497 . . . . . 6 (((𝜑𝑎 ∈ ℝ) ∧ (∀𝑏𝐴 ¬ 𝑏 < 𝑎 ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏))) → 𝑎 ∈ ℝ)
30 simprl 498 . . . . . 6 (((𝜑𝑎 ∈ ℝ) ∧ (∀𝑏𝐴 ¬ 𝑏 < 𝑎 ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏))) → ∀𝑏𝐴 ¬ 𝑏 < 𝑎)
31 elrabi 2759 . . . . . . . . . . . 12 (𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → 𝑦 ∈ ℝ)
32 negeq 7612 . . . . . . . . . . . . . . 15 (𝑤 = 𝑦 → -𝑤 = -𝑦)
3332eleq1d 2153 . . . . . . . . . . . . . 14 (𝑤 = 𝑦 → (-𝑤𝐴 ↔ -𝑦𝐴))
3433elrab3 2763 . . . . . . . . . . . . 13 (𝑦 ∈ ℝ → (𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ↔ -𝑦𝐴))
3534biimpd 142 . . . . . . . . . . . 12 (𝑦 ∈ ℝ → (𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → -𝑦𝐴))
3631, 35mpcom 36 . . . . . . . . . . 11 (𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → -𝑦𝐴)
37 breq1 3823 . . . . . . . . . . . . 13 (𝑏 = -𝑦 → (𝑏 < 𝑎 ↔ -𝑦 < 𝑎))
3837notbid 625 . . . . . . . . . . . 12 (𝑏 = -𝑦 → (¬ 𝑏 < 𝑎 ↔ ¬ -𝑦 < 𝑎))
3938rspcv 2711 . . . . . . . . . . 11 (-𝑦𝐴 → (∀𝑏𝐴 ¬ 𝑏 < 𝑎 → ¬ -𝑦 < 𝑎))
4036, 39syl 14 . . . . . . . . . 10 (𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} → (∀𝑏𝐴 ¬ 𝑏 < 𝑎 → ¬ -𝑦 < 𝑎))
4140adantr 270 . . . . . . . . 9 ((𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ∧ 𝑎 ∈ ℝ) → (∀𝑏𝐴 ¬ 𝑏 < 𝑎 → ¬ -𝑦 < 𝑎))
42 ltnegcon1 7878 . . . . . . . . . . . 12 ((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (-𝑎 < 𝑦 ↔ -𝑦 < 𝑎))
4342ancoms 264 . . . . . . . . . . 11 ((𝑦 ∈ ℝ ∧ 𝑎 ∈ ℝ) → (-𝑎 < 𝑦 ↔ -𝑦 < 𝑎))
4443notbid 625 . . . . . . . . . 10 ((𝑦 ∈ ℝ ∧ 𝑎 ∈ ℝ) → (¬ -𝑎 < 𝑦 ↔ ¬ -𝑦 < 𝑎))
4531, 44sylan 277 . . . . . . . . 9 ((𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ∧ 𝑎 ∈ ℝ) → (¬ -𝑎 < 𝑦 ↔ ¬ -𝑦 < 𝑎))
4641, 45sylibrd 167 . . . . . . . 8 ((𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ∧ 𝑎 ∈ ℝ) → (∀𝑏𝐴 ¬ 𝑏 < 𝑎 → ¬ -𝑎 < 𝑦))
4746ancoms 264 . . . . . . 7 ((𝑎 ∈ ℝ ∧ 𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}) → (∀𝑏𝐴 ¬ 𝑏 < 𝑎 → ¬ -𝑎 < 𝑦))
4847ralrimdva 2449 . . . . . 6 (𝑎 ∈ ℝ → (∀𝑏𝐴 ¬ 𝑏 < 𝑎 → ∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ -𝑎 < 𝑦))
4929, 30, 48sylc 61 . . . . 5 (((𝜑𝑎 ∈ ℝ) ∧ (∀𝑏𝐴 ¬ 𝑏 < 𝑎 ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏))) → ∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ -𝑎 < 𝑦)
50 nfv 1464 . . . . . . . . . . . 12 𝑐(𝜑𝑎 ∈ ℝ)
51 nfcv 2225 . . . . . . . . . . . . 13 𝑐
52 nfv 1464 . . . . . . . . . . . . . 14 𝑐 𝑎 < 𝑏
53 nfre1 2415 . . . . . . . . . . . . . 14 𝑐𝑐𝐴 𝑐 < 𝑏
5452, 53nfim 1507 . . . . . . . . . . . . 13 𝑐(𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)
5551, 54nfralya 2412 . . . . . . . . . . . 12 𝑐𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)
5650, 55nfan 1500 . . . . . . . . . . 11 𝑐((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏))
57 nfv 1464 . . . . . . . . . . 11 𝑐 𝑦 ∈ ℝ
5856, 57nfan 1500 . . . . . . . . . 10 𝑐(((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) ∧ 𝑦 ∈ ℝ)
59 nfv 1464 . . . . . . . . . 10 𝑐 𝑦 < -𝑎
6058, 59nfan 1500 . . . . . . . . 9 𝑐((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < -𝑎)
61 simplr 497 . . . . . . . . . . . . 13 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < -𝑎) ∧ 𝑐𝐴) ∧ 𝑐 < -𝑦) → 𝑐𝐴)
62 infsupneg.ss . . . . . . . . . . . . . . 15 (𝜑𝐴 ⊆ ℝ)
6362sseld 3013 . . . . . . . . . . . . . 14 (𝜑 → (𝑐𝐴𝑐 ∈ ℝ))
6463ad6antr 482 . . . . . . . . . . . . 13 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < -𝑎) ∧ 𝑐𝐴) ∧ 𝑐 < -𝑦) → (𝑐𝐴𝑐 ∈ ℝ))
6561, 64mpd 13 . . . . . . . . . . . 12 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < -𝑎) ∧ 𝑐𝐴) ∧ 𝑐 < -𝑦) → 𝑐 ∈ ℝ)
6665renegcld 7795 . . . . . . . . . . 11 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < -𝑎) ∧ 𝑐𝐴) ∧ 𝑐 < -𝑦) → -𝑐 ∈ ℝ)
6765recnd 7453 . . . . . . . . . . . . 13 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < -𝑎) ∧ 𝑐𝐴) ∧ 𝑐 < -𝑦) → 𝑐 ∈ ℂ)
6867negnegd 7721 . . . . . . . . . . . 12 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < -𝑎) ∧ 𝑐𝐴) ∧ 𝑐 < -𝑦) → --𝑐 = 𝑐)
6968, 61eqeltrd 2161 . . . . . . . . . . 11 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < -𝑎) ∧ 𝑐𝐴) ∧ 𝑐 < -𝑦) → --𝑐𝐴)
70 negeq 7612 . . . . . . . . . . . . 13 (𝑤 = -𝑐 → -𝑤 = --𝑐)
7170eleq1d 2153 . . . . . . . . . . . 12 (𝑤 = -𝑐 → (-𝑤𝐴 ↔ --𝑐𝐴))
7271elrab 2762 . . . . . . . . . . 11 (-𝑐 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ↔ (-𝑐 ∈ ℝ ∧ --𝑐𝐴))
7366, 69, 72sylanbrc 408 . . . . . . . . . 10 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < -𝑎) ∧ 𝑐𝐴) ∧ 𝑐 < -𝑦) → -𝑐 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴})
74 simp-4r 509 . . . . . . . . . . 11 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < -𝑎) ∧ 𝑐𝐴) ∧ 𝑐 < -𝑦) → 𝑦 ∈ ℝ)
75 simpr 108 . . . . . . . . . . 11 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < -𝑎) ∧ 𝑐𝐴) ∧ 𝑐 < -𝑦) → 𝑐 < -𝑦)
7665, 74, 75ltnegcon2d 7937 . . . . . . . . . 10 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < -𝑎) ∧ 𝑐𝐴) ∧ 𝑐 < -𝑦) → 𝑦 < -𝑐)
77 breq2 3824 . . . . . . . . . . 11 (𝑧 = -𝑐 → (𝑦 < 𝑧𝑦 < -𝑐))
7877rspcev 2715 . . . . . . . . . 10 ((-𝑐 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ∧ 𝑦 < -𝑐) → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑦 < 𝑧)
7973, 76, 78syl2anc 403 . . . . . . . . 9 (((((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < -𝑎) ∧ 𝑐𝐴) ∧ 𝑐 < -𝑦) → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑦 < 𝑧)
80 simpllr 501 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) ∧ 𝑦 ∈ ℝ) → 𝑎 ∈ ℝ)
81 simpr 108 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) ∧ 𝑦 ∈ ℝ) → 𝑦 ∈ ℝ)
82 simplr 497 . . . . . . . . . . 11 ((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) ∧ 𝑦 ∈ ℝ) → ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏))
8380, 81, 82jca31 302 . . . . . . . . . 10 ((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) ∧ 𝑦 ∈ ℝ) → ((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)))
84 ltnegcon2 7879 . . . . . . . . . . . . . 14 ((𝑦 ∈ ℝ ∧ 𝑎 ∈ ℝ) → (𝑦 < -𝑎𝑎 < -𝑦))
8584ancoms 264 . . . . . . . . . . . . 13 ((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑦 < -𝑎𝑎 < -𝑦))
8685adantr 270 . . . . . . . . . . . 12 (((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) → (𝑦 < -𝑎𝑎 < -𝑦))
87 renegcl 7680 . . . . . . . . . . . . . . 15 (𝑦 ∈ ℝ → -𝑦 ∈ ℝ)
88 breq2 3824 . . . . . . . . . . . . . . . . 17 (𝑏 = -𝑦 → (𝑎 < 𝑏𝑎 < -𝑦))
89 breq2 3824 . . . . . . . . . . . . . . . . . 18 (𝑏 = -𝑦 → (𝑐 < 𝑏𝑐 < -𝑦))
9089rexbidv 2377 . . . . . . . . . . . . . . . . 17 (𝑏 = -𝑦 → (∃𝑐𝐴 𝑐 < 𝑏 ↔ ∃𝑐𝐴 𝑐 < -𝑦))
9188, 90imbi12d 232 . . . . . . . . . . . . . . . 16 (𝑏 = -𝑦 → ((𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏) ↔ (𝑎 < -𝑦 → ∃𝑐𝐴 𝑐 < -𝑦)))
9291rspcv 2711 . . . . . . . . . . . . . . 15 (-𝑦 ∈ ℝ → (∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏) → (𝑎 < -𝑦 → ∃𝑐𝐴 𝑐 < -𝑦)))
9387, 92syl 14 . . . . . . . . . . . . . 14 (𝑦 ∈ ℝ → (∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏) → (𝑎 < -𝑦 → ∃𝑐𝐴 𝑐 < -𝑦)))
9493adantl 271 . . . . . . . . . . . . 13 ((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏) → (𝑎 < -𝑦 → ∃𝑐𝐴 𝑐 < -𝑦)))
9594imp 122 . . . . . . . . . . . 12 (((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) → (𝑎 < -𝑦 → ∃𝑐𝐴 𝑐 < -𝑦))
9686, 95sylbid 148 . . . . . . . . . . 11 (((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) → (𝑦 < -𝑎 → ∃𝑐𝐴 𝑐 < -𝑦))
9796imp 122 . . . . . . . . . 10 ((((𝑎 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) ∧ 𝑦 < -𝑎) → ∃𝑐𝐴 𝑐 < -𝑦)
9883, 97sylan 277 . . . . . . . . 9 (((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < -𝑎) → ∃𝑐𝐴 𝑐 < -𝑦)
9960, 79, 98r19.29af 2505 . . . . . . . 8 (((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) ∧ 𝑦 ∈ ℝ) ∧ 𝑦 < -𝑎) → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑦 < 𝑧)
10099ex 113 . . . . . . 7 ((((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) ∧ 𝑦 ∈ ℝ) → (𝑦 < -𝑎 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑦 < 𝑧))
101100ralrimiva 2442 . . . . . 6 (((𝜑𝑎 ∈ ℝ) ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) → ∀𝑦 ∈ ℝ (𝑦 < -𝑎 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑦 < 𝑧))
102101adantrl 462 . . . . 5 (((𝜑𝑎 ∈ ℝ) ∧ (∀𝑏𝐴 ¬ 𝑏 < 𝑎 ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏))) → ∀𝑦 ∈ ℝ (𝑦 < -𝑎 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑦 < 𝑧))
103 breq1 3823 . . . . . . . . 9 (𝑥 = -𝑎 → (𝑥 < 𝑦 ↔ -𝑎 < 𝑦))
104103notbid 625 . . . . . . . 8 (𝑥 = -𝑎 → (¬ 𝑥 < 𝑦 ↔ ¬ -𝑎 < 𝑦))
105104ralbidv 2376 . . . . . . 7 (𝑥 = -𝑎 → (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑥 < 𝑦 ↔ ∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ -𝑎 < 𝑦))
106 breq2 3824 . . . . . . . . 9 (𝑥 = -𝑎 → (𝑦 < 𝑥𝑦 < -𝑎))
107106imbi1d 229 . . . . . . . 8 (𝑥 = -𝑎 → ((𝑦 < 𝑥 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑦 < 𝑧) ↔ (𝑦 < -𝑎 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑦 < 𝑧)))
108107ralbidv 2376 . . . . . . 7 (𝑥 = -𝑎 → (∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑦 < 𝑧) ↔ ∀𝑦 ∈ ℝ (𝑦 < -𝑎 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑦 < 𝑧)))
109105, 108anbi12d 457 . . . . . 6 (𝑥 = -𝑎 → ((∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑦 < 𝑧)) ↔ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ -𝑎 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < -𝑎 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑦 < 𝑧))))
110109rspcev 2715 . . . . 5 ((-𝑎 ∈ ℝ ∧ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ -𝑎 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < -𝑎 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑦 < 𝑧))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑦 < 𝑧)))
11128, 49, 102, 110syl12anc 1170 . . . 4 (((𝜑𝑎 ∈ ℝ) ∧ (∀𝑏𝐴 ¬ 𝑏 < 𝑎 ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏))) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑦 < 𝑧)))
112111ex 113 . . 3 ((𝜑𝑎 ∈ ℝ) → ((∀𝑏𝐴 ¬ 𝑏 < 𝑎 ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑦 < 𝑧))))
113112rexlimdva 2485 . 2 (𝜑 → (∃𝑎 ∈ ℝ (∀𝑏𝐴 ¬ 𝑏 < 𝑎 ∧ ∀𝑏 ∈ ℝ (𝑎 < 𝑏 → ∃𝑐𝐴 𝑐 < 𝑏)) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑦 < 𝑧))))
11426, 113mpd 13 1 (𝜑 → ∃𝑥 ∈ ℝ (∀𝑦 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴} ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ {𝑤 ∈ ℝ ∣ -𝑤𝐴}𝑦 < 𝑧)))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103   = wceq 1287  wcel 1436  wral 2355  wrex 2356  {crab 2359  wss 2988   class class class wbr 3820  cr 7286   < clt 7459  -cneg 7591
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-13 1447  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3931  ax-pow 3983  ax-pr 4009  ax-un 4233  ax-setind 4325  ax-cnex 7373  ax-resscn 7374  ax-1cn 7375  ax-1re 7376  ax-icn 7377  ax-addcl 7378  ax-addrcl 7379  ax-mulcl 7380  ax-addcom 7382  ax-addass 7384  ax-distr 7386  ax-i2m1 7387  ax-0id 7390  ax-rnegex 7391  ax-cnre 7393  ax-pre-ltadd 7398
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-fal 1293  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-ne 2252  df-nel 2347  df-ral 2360  df-rex 2361  df-reu 2362  df-rab 2364  df-v 2617  df-sbc 2830  df-dif 2990  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-opab 3875  df-id 4093  df-xp 4416  df-rel 4417  df-cnv 4418  df-co 4419  df-dm 4420  df-iota 4943  df-fun 4980  df-fv 4986  df-riota 5563  df-ov 5610  df-oprab 5611  df-mpt2 5612  df-pnf 7461  df-mnf 7462  df-ltxr 7464  df-sub 7592  df-neg 7593
This theorem is referenced by:  infssuzcldc  10814
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