elioore - Intuitionistic Logic Explorer

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Theorem elioore 9688

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 9686	. 2 ⊢ (𝐴 ∈ (𝐵(,)𝐶) ↔ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) ∧ (𝐵 < 𝐴 ∧ 𝐴 < 𝐶)))
2		3ancomb 970	. . 3 ⊢ ((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^) ↔ (𝐵 ∈ ℝ^ ∧ 𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*))
3		xrre2 9597	. . 3 ⊢ (((𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^*) ∧ (𝐵 < 𝐴 ∧ 𝐴 < 𝐶)) → 𝐴 ∈ ℝ)
4	2, 3	sylanb 282	. 2 ⊢ (((𝐵 ∈ ℝ^* ∧ 𝐶 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) ∧ (𝐵 < 𝐴 ∧ 𝐴 < 𝐶)) → 𝐴 ∈ ℝ)
5	1, 4	sylbi 120	1 ⊢ (𝐴 ∈ (𝐵(,)𝐶) → 𝐴 ∈ ℝ)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 962 ∈ wcel 1480 class class class wbr 3924 (class class class)co 5767 ℝcr 7612 ℝ^*cxr 7792 < clt 7793 (,)cioo 9664

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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727

This theorem depends on definitions: df-bi 116 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-po 4213 df-iso 4214 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-iota 5083 df-fun 5120 df-fv 5126 df-ov 5770 df-oprab 5771 df-mpo 5772 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-ioo 9668

This theorem is referenced by: iooval2 9691 elioo4g 9710 ioossre 9711 zltaddlt1le 9782 tgioo 12704 ivthinc 12779 sin0pilem1 12851 sin0pilem2 12852 pilem3 12853 pire 12856 sinq34lt0t 12901 cosq14gt0 12902 cosq23lt0 12903 coseq0q4123 12904 tanrpcl 12907 tangtx 12908 cos02pilt1 12921

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