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Theorem lbicc2 11404
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 988 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  RR* )
2 xrleid 11130 . . 3  |-  ( A  e.  RR*  ->  A  <_  A )
323ad2ant1 1009 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  <_  A )
4 simp3 990 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  <_  B )
5 elicc1 11347 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  e.  ( A [,] B )  <->  ( A  e.  RR*  /\  A  <_  A  /\  A  <_  B
) ) )
653adant3 1008 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( A  e.  ( A [,] B )  <->  ( A  e.  RR*  /\  A  <_  A  /\  A  <_  B
) ) )
71, 3, 4, 6mpbir3and 1171 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  ( A [,] B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965    e. wcel 1756   class class class wbr 4295  (class class class)co 6094   RR*cxr 9420    <_ cle 9422   [,]cicc 11306
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4416  ax-nul 4424  ax-pow 4473  ax-pr 4534  ax-un 6375  ax-cnex 9341  ax-resscn 9342  ax-pre-lttri 9359  ax-pre-lttrn 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2571  df-ne 2611  df-nel 2612  df-ral 2723  df-rex 2724  df-rab 2727  df-v 2977  df-sbc 3190  df-csb 3292  df-dif 3334  df-un 3336  df-in 3338  df-ss 3345  df-nul 3641  df-if 3795  df-pw 3865  df-sn 3881  df-pr 3883  df-op 3887  df-uni 4095  df-br 4296  df-opab 4354  df-mpt 4355  df-id 4639  df-po 4644  df-so 4645  df-xp 4849  df-rel 4850  df-cnv 4851  df-co 4852  df-dm 4853  df-rn 4854  df-res 4855  df-ima 4856  df-iota 5384  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6097  df-oprab 6098  df-mpt2 6099  df-er 7104  df-en 7314  df-dom 7315  df-sdom 7316  df-pnf 9423  df-mnf 9424  df-xr 9425  df-ltxr 9426  df-le 9427  df-icc 11310
This theorem is referenced by:  icccmplem1  20402  reconnlem2  20407  oprpiece1res1  20526  pcoass  20599  ivthlem1  20938  ivth2  20942  ivthle  20943  ivthle2  20944  evthicc  20946  ovolicc2lem5  21007  dyadmaxlem  21080  rolle  21465  cmvth  21466  mvth  21467  dvlip  21468  c1liplem1  21471  dveq0  21475  dvgt0lem1  21477  lhop1lem  21488  dvcnvrelem1  21492  dvcvx  21495  dvfsumle  21496  dvfsumge  21497  dvfsumabs  21498  dvfsumlem2  21502  ftc2  21519  ftc2ditglem  21520  itgparts  21522  itgsubstlem  21523  taylfval  21827  tayl0  21830  efcvx  21917  pige3  21982  logccv  22111  loglesqr  22199  xrge0infss  26056  eliccioo  26109  oms0  26713  cvmliftlem6  27182  cvmliftlem8  27184  cvmliftlem9  27185  cvmliftlem10  27186  cvmliftlem13  27188  ftc2nc  28479  areacirc  28492  ivthALT  28533  itgpowd  29593  itgsin0pilem1  29793
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