elioore - Intuitionistic Logic Explorer

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Theorem elioore 8935

Description: A member of an open interval of reals is a real. (Contributed by NM, 17-Aug-2008.) (Revised by Mario Carneiro, 3-Nov-2013.)

**Assertion**
Ref	Expression
elioore	⊢ (𝐴 ∈ (𝐵(,)𝐶) → 𝐴 ∈ ℝ)

**Proof of Theorem elioore**
Step	Hyp	Ref	Expression
1		elioo3g 8933	. 2 ⊢ (𝐴 ∈ (𝐵(,)𝐶) ↔ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) ∧ (𝐵 < 𝐴 ∧ 𝐴 < 𝐶)))
2		3ancomb 927	. . 3 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^) ↔ (𝐵 ∈ ℝ^ ∧ 𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*))
3		xrre2 8888	. . 3 ⊢ (((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐵 < 𝐴 ∧ 𝐴 < 𝐶)) → 𝐴 ∈ ℝ)
4	2, 3	sylanb 278	. 2 ⊢ (((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) ∧ (𝐵 < 𝐴 ∧ 𝐴 < 𝐶)) → 𝐴 ∈ ℝ)
5	1, 4	sylbi 119	1 ⊢ (𝐴 ∈ (𝐵(,)𝐶) → 𝐴 ∈ ℝ)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 102 ∧ w3a 919 ∈ wcel 1433 class class class wbr 3785 (class class class)co 5532 ℝcr 6980 ℝ^*cxr 7152 < clt 7153 (,)cioo 8911

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-setind 4280 ax-cnex 7067 ax-resscn 7068 ax-pre-ltirr 7088 ax-pre-ltwlin 7089 ax-pre-lttrn 7090

This theorem depends on definitions: df-bi 115 df-3or 920 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ne 2246 df-nel 2340 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-id 4048 df-po 4051 df-iso 4052 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-iota 4887 df-fun 4924 df-fv 4930 df-ov 5535 df-oprab 5536 df-mpt2 5537 df-pnf 7155 df-mnf 7156 df-xr 7157 df-ltxr 7158 df-le 7159 df-ioo 8915

This theorem is referenced by: iooval2 8938 elioo4g 8957 ioossre 8958 zltaddlt1le 9028

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