elioo4g - Metamath Proof Explorer

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Theorem elioo4g 13484

Description: Membership in an open interval of extended reals. (Contributed by NM, 8-Jun-2007.) (Revised by Mario Carneiro, 28-Apr-2015.)

**Assertion**
Ref	Expression
elioo4g	⊢ (𝐶 ∈ (𝐴(,)𝐵) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) ∧ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))

**Proof of Theorem elioo4g**
Step	Hyp	Ref	Expression
1		eliooxr 13482	. . . . 5 ⊢ (𝐶 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
2		elioore 13453	. . . . 5 ⊢ (𝐶 ∈ (𝐴(,)𝐵) → 𝐶 ∈ ℝ)
3	1, 2	jca 521	. . . 4 ⊢ (𝐶 ∈ (𝐴(,)𝐵) → ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 ∈ ℝ))
4		df-3an 1105	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 ∈ ℝ))
5	3, 4	sylibr 237	. . 3 ⊢ (𝐶 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ))
6		eliooord 13483	. . 3 ⊢ (𝐶 ∈ (𝐴(,)𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))
7	5, 6	jca 521	. 2 ⊢ (𝐶 ∈ (𝐴(,)𝐵) → ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) ∧ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
8		rexr 11304	. . . . 5 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ^*)
9	8	3anim3i 1172	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*))
10	9	anim1i 627	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) ∧ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) → ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
11		elioo3g 13452	. . 3 ⊢ (𝐶 ∈ (𝐴(,)𝐵) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
12	10, 11	sylibr 237	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) ∧ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) → 𝐶 ∈ (𝐴(,)𝐵))
13	7, 12	impbii 212	1 ⊢ (𝐶 ∈ (𝐴(,)𝐵) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) ∧ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ↔ wb 209 ∧ wa 401 ∧ w3a 1103 ∈ wcel 2145 class class class wbr 5103 (class class class)co 7416 ℝcr 11148 ℝ^*cxr 11291 < clt 11292 (,)cioo 13423

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pow 5330 ax-pr 5398 ax-un 7742 ax-cnex 11205 ax-resscn 11206 ax-pre-lttri 11223 ax-pre-lttrn 11224

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-po 5563 df-so 5564 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-iota 6491 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-ov 7419 df-oprab 7420 df-mpo 7421 df-1st 7992 df-2nd 7993 df-er 8703 df-en 8960 df-dom 8961 df-sdom 8962 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-ioo 13427

This theorem is used by: cncfiooicclem1 46786 fourierdlem89 47088 fourierdlem90 47089 fourierdlem91 47090 fourierdlem100 47099

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