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Theorem icossxr 11530
Description: A closed-below, open-above interval is a subset of the extended reals. (Contributed by FL, 29-May-2014.) (Revised by Mario Carneiro, 4-Jul-2014.)
Assertion
Ref Expression
icossxr  |-  ( A [,) B )  C_  RR*

Proof of Theorem icossxr
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ico 11456 . 2  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
21ixxssxr 11462 1  |-  ( A [,) B )  C_  RR*
Colors of variables: wff setvar class
Syntax hints:    C_ wss 3389  (class class class)co 6196   RR*cxr 9538    < clt 9539    <_ cle 9540   [,)cico 11452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-1st 6699  df-2nd 6700  df-xr 9543  df-ico 11456
This theorem is referenced by:  leordtvallem2  19798  leordtval2  19799  nmoffn  21303  nmofval  21306  nmogelb  21308  nmolb  21309  nmof  21311  icopnfhmeo  21528  elovolm  21971  ovolmge0  21973  ovolgelb  21976  ovollb2lem  21984  ovoliunlem1  21998  ovoliunlem2  21999  ovolscalem1  22009  ovolicc1  22012  ioombl1lem2  22054  ioombl1lem4  22056  uniioovol  22073  uniiccvol  22074  uniioombllem1  22075  uniioombllem2  22077  uniioombllem3  22079  uniioombllem6  22082  esumpfinvallem  28222  esummulc1  28229  esummulc2  28230  mblfinlem3  30218  mblfinlem4  30219  ismblfin  30220  itg2gt0cn  30236
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