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Theorem elioore 11318
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  e.  ( B (,) C )  ->  A  e.  RR )

Proof of Theorem elioore
StepHypRef Expression
1 elioo3g 11317 . 2  |-  ( A  e.  ( B (,) C )  <->  ( ( B  e.  RR*  /\  C  e.  RR*  /\  A  e. 
RR* )  /\  ( B  <  A  /\  A  <  C ) ) )
2 3ancomb 967 . . 3  |-  ( ( B  e.  RR*  /\  C  e.  RR*  /\  A  e. 
RR* )  <->  ( B  e.  RR*  /\  A  e. 
RR*  /\  C  e.  RR* ) )
3 xrre2 11130 . . 3  |-  ( ( ( B  e.  RR*  /\  A  e.  RR*  /\  C  e.  RR* )  /\  ( B  <  A  /\  A  <  C ) )  ->  A  e.  RR )
42, 3sylanb 469 . 2  |-  ( ( ( B  e.  RR*  /\  C  e.  RR*  /\  A  e.  RR* )  /\  ( B  <  A  /\  A  <  C ) )  ->  A  e.  RR )
51, 4sylbi 195 1  |-  ( A  e.  ( B (,) C )  ->  A  e.  RR )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 958    e. wcel 1755   class class class wbr 4280  (class class class)co 6080   RRcr 9269   RR*cxr 9405    < clt 9406   (,)cioo 11288
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9326  ax-resscn 9327  ax-pre-lttri 9344  ax-pre-lttrn 9345
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-po 4628  df-so 4629  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-1st 6566  df-2nd 6567  df-er 7089  df-en 7299  df-dom 7300  df-sdom 7301  df-pnf 9408  df-mnf 9409  df-xr 9410  df-ltxr 9411  df-le 9412  df-ioo 11292
This theorem is referenced by:  iooval2  11321  elioo4g  11344  ioossre  11345  tgioo  20215  zcld  20232  ioorcl2  20894  lhop2  21329  dvcvx  21334  pilem2  21802  pilem3  21803  pire  21806  tanrpcl  21851  tangtx  21852  tanabsge  21853  sinq34lt0t  21856  cosq14gt0  21857  sineq0  21868  cosne0  21871  tanord  21879  divlogrlim  21965  logno1  21966  logccv  21993  angpieqvd  22111  asinsin  22172  reasinsin  22176  scvxcvx  22264  basellem3  22305  basellem8  22310  vmalogdivsum2  22672  vmalogdivsum  22673  2vmadivsumlem  22674  selberg3lem1  22691  selberg3  22693  selberg4lem1  22694  selberg4  22695  selberg3r  22703  selberg4r  22704  selberg34r  22705  pntrlog2bndlem1  22711  pntrlog2bndlem2  22712  pntrlog2bndlem3  22713  pntrlog2bndlem4  22714  pntrlog2bndlem5  22715  pntrlog2bndlem6a  22716  pntrlog2bndlem6  22717  pntpbnd  22722  pntibndlem3  22726  pntibnd  22727  tan2h  28268  dvtanlem  28285  itg2gt0cn  28291  itggt0cn  28308  ftc1cnnclem  28309  ftc1cnnc  28310  ftc1anclem7  28317  ftc1anclem8  28318  ftc1anc  28319  dvasin  28324  areacirclem1  28328  areacirc  28333  wallispilem1  29706
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