iooneg - Metamath Proof Explorer

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Theorem iooneg 12856

Description: Membership in a negated open real interval. (Contributed by Paul Chapman, 26-Nov-2007.)

**Assertion**
Ref	Expression
iooneg	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 ∈ (𝐴(,)𝐵) ↔ -𝐶 ∈ (-𝐵(,)-𝐴)))

**Proof of Theorem iooneg**
Step	Hyp	Ref	Expression
1		ltneg 11139	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐶 ↔ -𝐶 < -𝐴))
2	1	3adant2 1127	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐴 < 𝐶 ↔ -𝐶 < -𝐴))
3		ltneg 11139	. . . . . 6 ⊢ ((𝐶 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐶 < 𝐵 ↔ -𝐵 < -𝐶))
4	3	ancoms 461	. . . . 5 ⊢ ((𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 < 𝐵 ↔ -𝐵 < -𝐶))
5	4	3adant1 1126	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 < 𝐵 ↔ -𝐵 < -𝐶))
6	2, 5	anbi12d 632	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ↔ (-𝐶 < -𝐴 ∧ -𝐵 < -𝐶)))
7	6	biancomd 466	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → ((𝐴 < 𝐶 ∧ 𝐶 < 𝐵) ↔ (-𝐵 < -𝐶 ∧ -𝐶 < -𝐴)))
8		rexr 10686	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℝ^*)
9		rexr 10686	. . 3 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
10		rexr 10686	. . 3 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ^*)
11		elioo5 12793	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
12	8, 9, 10, 11	syl3an 1156	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 ∈ (𝐴(,)𝐵) ↔ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
13		renegcl 10948	. . . 4 ⊢ (𝐵 ∈ ℝ → -𝐵 ∈ ℝ)
14		renegcl 10948	. . . 4 ⊢ (𝐴 ∈ ℝ → -𝐴 ∈ ℝ)
15		renegcl 10948	. . . 4 ⊢ (𝐶 ∈ ℝ → -𝐶 ∈ ℝ)
16		rexr 10686	. . . . 5 ⊢ (-𝐵 ∈ ℝ → -𝐵 ∈ ℝ^*)
17		rexr 10686	. . . . 5 ⊢ (-𝐴 ∈ ℝ → -𝐴 ∈ ℝ^*)
18		rexr 10686	. . . . 5 ⊢ (-𝐶 ∈ ℝ → -𝐶 ∈ ℝ^*)
19		elioo5 12793	. . . . 5 ⊢ ((-𝐵 ∈ ℝ^* ∧ -𝐴 ∈ ℝ^* ∧ -𝐶 ∈ ℝ^*) → (-𝐶 ∈ (-𝐵(,)-𝐴) ↔ (-𝐵 < -𝐶 ∧ -𝐶 < -𝐴)))
20	16, 17, 18, 19	syl3an 1156	. . . 4 ⊢ ((-𝐵 ∈ ℝ ∧ -𝐴 ∈ ℝ ∧ -𝐶 ∈ ℝ) → (-𝐶 ∈ (-𝐵(,)-𝐴) ↔ (-𝐵 < -𝐶 ∧ -𝐶 < -𝐴)))
21	13, 14, 15, 20	syl3an 1156	. . 3 ⊢ ((𝐵 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (-𝐶 ∈ (-𝐵(,)-𝐴) ↔ (-𝐵 < -𝐶 ∧ -𝐶 < -𝐴)))
22	21	3com12 1119	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (-𝐶 ∈ (-𝐵(,)-𝐴) ↔ (-𝐵 < -𝐶 ∧ -𝐶 < -𝐴)))
23	7, 12, 22	3bitr4d 313	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝐶 ∈ ℝ) → (𝐶 ∈ (𝐴(,)𝐵) ↔ -𝐶 ∈ (-𝐵(,)-𝐴)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 ∈ wcel 2110 class class class wbr 5065 (class class class)co 7155 ℝcr 10535 ℝ^*cxr 10673 < clt 10674 -cneg 10870 (,)cioo 12737

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-po 5473 df-so 5474 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-ioo 12741

This theorem is referenced by: lhop2 24611 asinsin 25469 atanlogsub 25493 atanbnd 25503

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