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Theorem lbicc2 11632
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 996 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  e.  RR* )
2 xrleid 11352 . . 3  |-  ( A  e.  RR*  ->  A  <_  A )
323ad2ant1 1017 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  <_  A )
4 simp3 998 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  A  <_  B )
5 elicc1 11569 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  e.  ( A [,] B )  <->  ( A  e.  RR*  /\  A  <_  A  /\  A  <_  B
) ) )
653adant3 1016 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( A  e.  ( A [,] B )  <->  ( A  e.  RR*  /\  A  <_  A  /\  A  <_  B
) ) )
71, 3, 4, 6mpbir3and 1179 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 973    e. wcel 1767   class class class wbr 4447  (class class class)co 6282   RR*cxr 9623    <_ cle 9625   [,]cicc 11528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-pre-lttri 9562  ax-pre-lttrn 9563
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-icc 11532
This theorem is referenced by:  icccmplem1  21062  reconnlem2  21067  oprpiece1res1  21186  pcoass  21259  ivthlem1  21598  ivth2  21602  ivthle  21603  ivthle2  21604  evthicc  21606  ovolicc2lem5  21667  dyadmaxlem  21741  rolle  22126  cmvth  22127  mvth  22128  dvlip  22129  c1liplem1  22132  dveq0  22136  dvgt0lem1  22138  lhop1lem  22149  dvcnvrelem1  22153  dvcvx  22156  dvfsumle  22157  dvfsumge  22158  dvfsumabs  22159  dvfsumlem2  22163  ftc2  22180  ftc2ditglem  22181  itgparts  22183  itgsubstlem  22184  taylfval  22488  tayl0  22491  efcvx  22578  pige3  22643  logccv  22772  loglesqrt  22860  xrge0infss  27248  eliccioo  27295  oms0  27906  cvmliftlem6  28375  cvmliftlem8  28377  cvmliftlem9  28378  cvmliftlem10  28379  cvmliftlem13  28381  ftc2nc  29676  areacirc  29689  ivthALT  29730  itgpowd  30787  iccintsng  31127  icccncfext  31226  cncfiooicclem1  31232  dvbdfbdioolem1  31258  itgsin0pilem1  31267  itgcoscmulx  31287  itgsincmulx  31292  fourierdlem20  31427  fourierdlem51  31458  fourierdlem54  31461  fourierdlem64  31471  fourierdlem73  31480  fourierdlem81  31488  fourierdlem92  31499  fourierdlem93  31500  fourierdlem102  31509  fourierdlem103  31510  fourierdlem104  31511  fourierdlem114  31521
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