MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  lbicc2 Structured version   Unicode version

Theorem lbicc2 11013
Description: The lower bound of a closed interval is a member of it. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by FL, 29-May-2014.) (Revised by Mario Carneiro, 9-Sep-2015.)
Assertion
Ref Expression
lbicc2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )

Proof of Theorem lbicc2
StepHypRef Expression
1 simp1 957 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  RR* )
2 xrleid 10743 . . 3  |-  ( A  e.  RR*  ->  A  <_  A )
323ad2ant1 978 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  <_  A )
4 simp3 959 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  <_  B )
5 elicc1 10960 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  e.  ( A [,] B )  <->  ( A  e.  RR*  /\  A  <_  A  /\  A  <_  B
) ) )
653adant3 977 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( A  e.  ( A [,] B )  <->  ( A  e.  RR*  /\  A  <_  A  /\  A  <_  B
) ) )
71, 3, 4, 6mpbir3and 1137 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936    e. wcel 1725   class class class wbr 4212  (class class class)co 6081   RR*cxr 9119    <_ cle 9121   [,]cicc 10919
This theorem is referenced by:  icccmplem1  18853  reconnlem2  18858  oprpiece1res1  18976  pcoass  19049  ivthlem1  19348  ivth2  19352  ivthle  19353  ivthle2  19354  evthicc  19356  ovolicc2lem5  19417  dyadmaxlem  19489  rolle  19874  cmvth  19875  mvth  19876  dvlip  19877  c1liplem1  19880  dveq0  19884  dvgt0lem1  19886  lhop1lem  19897  dvcnvrelem1  19901  dvcvx  19904  dvfsumle  19905  dvfsumge  19906  dvfsumabs  19907  dvfsumlem2  19911  ftc2  19928  ftc2ditglem  19929  itgparts  19931  itgsubstlem  19932  taylfval  20275  tayl0  20278  efcvx  20365  pige3  20425  logccv  20554  loglesqr  20642  eliccioo  24177  unitssxrge0  24298  xrge0mulc1cn  24327  esum0  24444  esumpinfval  24463  esummulc1  24471  cvmliftlem6  24977  cvmliftlem8  24979  cvmliftlem9  24980  cvmliftlem10  24981  cvmliftlem13  24983  ftc1anc  26288  ftc2nc  26289  areacirc  26297  ivthALT  26338  itgsin0pilem1  27720
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-pre-lttri 9064  ax-pre-lttrn 9065
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-icc 10923
  Copyright terms: Public domain W3C validator