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Mirrors > Home > ILE Home > Th. List > iooneg | GIF version |
Description: Membership in a negated open real interval. (Contributed by Paul Chapman, 26-Nov-2007.) |
Ref | Expression |
---|---|
iooneg | ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 ∈ (𝐴(,)𝐵) ↔ -𝐶 ∈ (-𝐵(,)-𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltneg 8227 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐶 ↔ -𝐶 < -𝐴)) | |
2 | 1 | 3adant2 1000 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐶 ↔ -𝐶 < -𝐴)) |
3 | ltneg 8227 | . . . . . 6 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 < 𝐵 ↔ -𝐵 < -𝐶)) | |
4 | 3 | ancoms 266 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 < 𝐵 ↔ -𝐵 < -𝐶)) |
5 | 4 | 3adant1 999 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 < 𝐵 ↔ -𝐵 < -𝐶)) |
6 | 2, 5 | anbi12d 464 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ↔ (-𝐶 < -𝐴 ∧ -𝐵 < -𝐶))) |
7 | ancom 264 | . . 3 ⊢ ((-𝐶 < -𝐴 ∧ -𝐵 < -𝐶) ↔ (-𝐵 < -𝐶 ∧ -𝐶 < -𝐴)) | |
8 | 6, 7 | syl6bb 195 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ↔ (-𝐵 < -𝐶 ∧ -𝐶 < -𝐴))) |
9 | rexr 7814 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
10 | rexr 7814 | . . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
11 | rexr 7814 | . . 3 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ*) | |
12 | elioo5 9719 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) | |
13 | 9, 10, 11, 12 | syl3an 1258 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) |
14 | renegcl 8026 | . . . 4 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ) | |
15 | renegcl 8026 | . . . 4 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
16 | renegcl 8026 | . . . 4 ⊢ (𝐶 ∈ ℝ → -𝐶 ∈ ℝ) | |
17 | rexr 7814 | . . . . 5 ⊢ (-𝐵 ∈ ℝ → -𝐵 ∈ ℝ*) | |
18 | rexr 7814 | . . . . 5 ⊢ (-𝐴 ∈ ℝ → -𝐴 ∈ ℝ*) | |
19 | rexr 7814 | . . . . 5 ⊢ (-𝐶 ∈ ℝ → -𝐶 ∈ ℝ*) | |
20 | elioo5 9719 | . . . . 5 ⊢ ((-𝐵 ∈ ℝ* ∧ -𝐴 ∈ ℝ* ∧ -𝐶 ∈ ℝ*) → (-𝐶 ∈ (-𝐵(,)-𝐴) ↔ (-𝐵 < -𝐶 ∧ -𝐶 < -𝐴))) | |
21 | 17, 18, 19, 20 | syl3an 1258 | . . . 4 ⊢ ((-𝐵 ∈ ℝ ∧ -𝐴 ∈ ℝ ∧ -𝐶 ∈ ℝ) → (-𝐶 ∈ (-𝐵(,)-𝐴) ↔ (-𝐵 < -𝐶 ∧ -𝐶 < -𝐴))) |
22 | 14, 15, 16, 21 | syl3an 1258 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (-𝐶 ∈ (-𝐵(,)-𝐴) ↔ (-𝐵 < -𝐶 ∧ -𝐶 < -𝐴))) |
23 | 22 | 3com12 1185 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (-𝐶 ∈ (-𝐵(,)-𝐴) ↔ (-𝐵 < -𝐶 ∧ -𝐶 < -𝐴))) |
24 | 8, 13, 23 | 3bitr4d 219 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 ∈ (𝐴(,)𝐵) ↔ -𝐶 ∈ (-𝐵(,)-𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 962 ∈ wcel 1480 class class class wbr 3929 (class class class)co 5774 ℝcr 7622 ℝ*cxr 7802 < clt 7803 -cneg 7937 (,)cioo 9674 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7714 ax-resscn 7715 ax-1cn 7716 ax-1re 7717 ax-icn 7718 ax-addcl 7719 ax-addrcl 7720 ax-mulcl 7721 ax-addcom 7723 ax-addass 7725 ax-distr 7727 ax-i2m1 7728 ax-0id 7731 ax-rnegex 7732 ax-cnre 7734 ax-pre-ltadd 7739 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-iota 5088 df-fun 5125 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-pnf 7805 df-mnf 7806 df-xr 7807 df-ltxr 7808 df-sub 7938 df-neg 7939 df-ioo 9678 |
This theorem is referenced by: (None) |
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