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Theorem elioore 11440
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 11439 . 2  |-  ( A  e.  ( B (,) C )  <->  ( ( B  e.  RR*  /\  C  e.  RR*  /\  A  e. 
RR* )  /\  ( B  <  A  /\  A  <  C ) ) )
2 3ancomb 974 . . 3  |-  ( ( B  e.  RR*  /\  C  e.  RR*  /\  A  e. 
RR* )  <->  ( B  e.  RR*  /\  A  e. 
RR*  /\  C  e.  RR* ) )
3 xrre2 11252 . . 3  |-  ( ( ( B  e.  RR*  /\  A  e.  RR*  /\  C  e.  RR* )  /\  ( B  <  A  /\  A  <  C ) )  ->  A  e.  RR )
42, 3sylanb 472 . 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 965    e. wcel 1758   class class class wbr 4399  (class class class)co 6199   RRcr 9391   RR*cxr 9527    < clt 9528   (,)cioo 11410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-pre-lttri 9466  ax-pre-lttrn 9467
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-po 4748  df-so 4749  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-1st 6686  df-2nd 6687  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-ioo 11414
This theorem is referenced by:  iooval2  11443  elioo4g  11466  ioossre  11467  tgioo  20504  zcld  20521  ioorcl2  21184  lhop2  21619  dvcvx  21624  pilem2  22049  pilem3  22050  pire  22053  tanrpcl  22098  tangtx  22099  tanabsge  22100  sinq34lt0t  22103  cosq14gt0  22104  sineq0  22115  cosne0  22118  tanord  22126  divlogrlim  22212  logno1  22213  logccv  22240  angpieqvd  22358  asinsin  22419  reasinsin  22423  scvxcvx  22511  basellem3  22552  basellem8  22557  vmalogdivsum2  22919  vmalogdivsum  22920  2vmadivsumlem  22921  selberg3lem1  22938  selberg3  22940  selberg4lem1  22941  selberg4  22942  selberg3r  22950  selberg4r  22951  selberg34r  22952  pntrlog2bndlem1  22958  pntrlog2bndlem2  22959  pntrlog2bndlem3  22960  pntrlog2bndlem4  22961  pntrlog2bndlem5  22962  pntrlog2bndlem6a  22963  pntrlog2bndlem6  22964  pntpbnd  22969  pntibndlem3  22973  pntibnd  22974  tan2h  28571  dvtanlem  28588  itg2gt0cn  28594  itggt0cn  28611  ftc1cnnclem  28612  ftc1cnnc  28613  ftc1anclem7  28620  ftc1anclem8  28621  ftc1anc  28622  dvasin  28627  areacirclem1  28631  areacirc  28636  wallispilem1  30007
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