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Mirrors > Home > ILE Home > Th. List > elioore | GIF version |
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.) |
Ref | Expression |
---|---|
elioore | ⊢ (𝐴 ∈ (𝐵(,)𝐶) → 𝐴 ∈ ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elioo3g 9723 | . 2 ⊢ (𝐴 ∈ (𝐵(,)𝐶) ↔ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) ∧ (𝐵 < 𝐴 ∧ 𝐴 < 𝐶))) | |
2 | 3ancomb 971 | . . 3 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐶 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) ↔ (𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝐶 ∈ ℝ*)) | |
3 | xrre2 9634 | . . 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 963 ∈ wcel 1481 class class class wbr 3937 (class class class)co 5782 ℝcr 7643 ℝ*cxr 7823 < clt 7824 (,)cioo 9701 |
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 604 ax-in2 605 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-cnex 7735 ax-resscn 7736 ax-pre-ltirr 7756 ax-pre-ltwlin 7757 ax-pre-lttrn 7758 |
This theorem depends on definitions: df-bi 116 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-nel 2405 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-id 4223 df-po 4226 df-iso 4227 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-iota 5096 df-fun 5133 df-fv 5139 df-ov 5785 df-oprab 5786 df-mpo 5787 df-pnf 7826 df-mnf 7827 df-xr 7828 df-ltxr 7829 df-le 7830 df-ioo 9705 |
This theorem is referenced by: iooval2 9728 elioo4g 9747 ioossre 9748 zltaddlt1le 9820 tgioo 12754 ivthinc 12829 reeff1oleme 12901 sin0pilem1 12910 sin0pilem2 12911 pilem3 12912 pire 12915 sinq34lt0t 12960 cosq14gt0 12961 cosq23lt0 12962 coseq0q4123 12963 tanrpcl 12966 tangtx 12967 cos02pilt1 12980 cos0pilt1 12981 ioocosf1o 12983 iooref1o 13426 |
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