elioo4g - Intuitionistic Logic Explorer

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Theorem elioo4g 9895

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 9888	. . . . 5 ⊢ (𝐶 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*))
2		elioore 9873	. . . . 5 ⊢ (𝐶 ∈ (𝐴(,)𝐵) → 𝐶 ∈ ℝ)
3	1, 2	jca 304	. . . 4 ⊢ (𝐶 ∈ (𝐴(,)𝐵) → ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 ∈ ℝ))
4		df-3an 976	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) ∧ 𝐶 ∈ ℝ))
5	3, 4	sylibr 133	. . 3 ⊢ (𝐶 ∈ (𝐴(,)𝐵) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ))
6		eliooord 9889	. . 3 ⊢ (𝐶 ∈ (𝐴(,)𝐵) → (𝐴 < 𝐶 ∧ 𝐶 < 𝐵))
7	5, 6	jca 304	. 2 ⊢ (𝐶 ∈ (𝐴(,)𝐵) → ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) ∧ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
8		rexr 7969	. . . . 5 ⊢ (𝐶 ∈ ℝ → 𝐶 ∈ ℝ^*)
9	8	3anim3i 1183	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) → (𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*))
10	9	anim1i 338	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) ∧ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) → ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
11		elioo3g 9871	. . 3 ⊢ (𝐶 ∈ (𝐴(,)𝐵) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))
12	10, 11	sylibr 133	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) ∧ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)) → 𝐶 ∈ (𝐴(,)𝐵))
13	7, 12	impbii 125	1 ⊢ (𝐶 ∈ (𝐴(,)𝐵) ↔ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ) ∧ (𝐴 < 𝐶 ∧ 𝐶 < 𝐵)))

Colors of variables: wff set class

Syntax hints: ∧ wa 103 ↔ wb 104 ∧ w3a 974 ∈ wcel 2142 class class class wbr 3990 (class class class)co 5857 ℝcr 7777 ℝ^*cxr 7957 < clt 7958 (,)cioo 9849

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 610 ax-in2 611 ax-io 705 ax-5 1441 ax-7 1442 ax-gen 1443 ax-ie1 1487 ax-ie2 1488 ax-8 1498 ax-10 1499 ax-11 1500 ax-i12 1501 ax-bndl 1503 ax-4 1504 ax-17 1520 ax-i9 1524 ax-ial 1528 ax-i5r 1529 ax-13 2144 ax-14 2145 ax-ext 2153 ax-sep 4108 ax-pow 4161 ax-pr 4195 ax-un 4419 ax-setind 4522 ax-cnex 7869 ax-resscn 7870 ax-pre-ltirr 7890 ax-pre-ltwlin 7891 ax-pre-lttrn 7892

This theorem depends on definitions: df-bi 116 df-3or 975 df-3an 976 df-tru 1352 df-fal 1355 df-nf 1455 df-sb 1757 df-eu 2023 df-mo 2024 df-clab 2158 df-cleq 2164 df-clel 2167 df-nfc 2302 df-ne 2342 df-nel 2437 df-ral 2454 df-rex 2455 df-rab 2458 df-v 2733 df-sbc 2957 df-dif 3124 df-un 3126 df-in 3128 df-ss 3135 df-pw 3569 df-sn 3590 df-pr 3591 df-op 3593 df-uni 3798 df-br 3991 df-opab 4052 df-id 4279 df-po 4282 df-iso 4283 df-xp 4618 df-rel 4619 df-cnv 4620 df-co 4621 df-dm 4622 df-iota 5162 df-fun 5202 df-fv 5208 df-ov 5860 df-oprab 5861 df-mpo 5862 df-pnf 7960 df-mnf 7961 df-xr 7962 df-ltxr 7963 df-le 7964 df-ioo 9853

This theorem is referenced by: (None)

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