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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 8450 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐶 ↔ -𝐶 < -𝐴)) | |
2 | 1 | 3adant2 1018 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐶 ↔ -𝐶 < -𝐴)) |
3 | ltneg 8450 | . . . . . 6 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 < 𝐵 ↔ -𝐵 < -𝐶)) | |
4 | 3 | ancoms 268 | . . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 < 𝐵 ↔ -𝐵 < -𝐶)) |
5 | 4 | 3adant1 1017 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 < 𝐵 ↔ -𝐵 < -𝐶)) |
6 | 2, 5 | anbi12d 473 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ↔ (-𝐶 < -𝐴 ∧ -𝐵 < -𝐶))) |
7 | ancom 266 | . . 3 ⊢ ((-𝐶 < -𝐴 ∧ -𝐵 < -𝐶) ↔ (-𝐵 < -𝐶 ∧ -𝐶 < -𝐴)) | |
8 | 6, 7 | bitrdi 196 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ↔ (-𝐵 < -𝐶 ∧ -𝐶 < -𝐴))) |
9 | rexr 8034 | . . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ*) | |
10 | rexr 8034 | . . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ*) | |
11 | rexr 8034 | . . 3 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ*) | |
12 | elioo5 9965 | . . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) | |
13 | 9, 10, 11, 12 | syl3an 1291 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))) |
14 | renegcl 8249 | . . . 4 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ) | |
15 | renegcl 8249 | . . . 4 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ) | |
16 | renegcl 8249 | . . . 4 ⊢ (𝐶 ∈ ℝ → -𝐶 ∈ ℝ) | |
17 | rexr 8034 | . . . . 5 ⊢ (-𝐵 ∈ ℝ → -𝐵 ∈ ℝ*) | |
18 | rexr 8034 | . . . . 5 ⊢ (-𝐴 ∈ ℝ → -𝐴 ∈ ℝ*) | |
19 | rexr 8034 | . . . . 5 ⊢ (-𝐶 ∈ ℝ → -𝐶 ∈ ℝ*) | |
20 | elioo5 9965 | . . . . 5 ⊢ ((-𝐵 ∈ ℝ* ∧ -𝐴 ∈ ℝ* ∧ -𝐶 ∈ ℝ*) → (-𝐶 ∈ (-𝐵(,)-𝐴) ↔ (-𝐵 < -𝐶 ∧ -𝐶 < -𝐴))) | |
21 | 17, 18, 19, 20 | syl3an 1291 | . . . 4 ⊢ ((-𝐵 ∈ ℝ ∧ -𝐴 ∈ ℝ ∧ -𝐶 ∈ ℝ) → (-𝐶 ∈ (-𝐵(,)-𝐴) ↔ (-𝐵 < -𝐶 ∧ -𝐶 < -𝐴))) |
22 | 14, 15, 16, 21 | syl3an 1291 | . . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (-𝐶 ∈ (-𝐵(,)-𝐴) ↔ (-𝐵 < -𝐶 ∧ -𝐶 < -𝐴))) |
23 | 22 | 3com12 1209 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (-𝐶 ∈ (-𝐵(,)-𝐴) ↔ (-𝐵 < -𝐶 ∧ -𝐶 < -𝐴))) |
24 | 8, 13, 23 | 3bitr4d 220 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 ∈ (𝐴(,)𝐵) ↔ -𝐶 ∈ (-𝐵(,)-𝐴))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 980 ∈ wcel 2160 class class class wbr 4018 (class class class)co 5897 ℝcr 7841 ℝ*cxr 8022 < clt 8023 -cneg 8160 (,)cioo 9920 |
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-xr 8027 df-ltxr 8028 df-sub 8161 df-neg 8162 df-ioo 9924 |
This theorem is referenced by: (None) |
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