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Theorem icossicc 26009
Description: A closed-below, open-above interval is a subset of its closure. (Contributed by Thierry Arnoux, 25-Oct-2016.)
Assertion
Ref Expression
icossicc  |-  ( A [,) B )  C_  ( A [,] B )

Proof of Theorem icossicc
Dummy variables  a 
b  w  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ico 11298 . 2  |-  [,)  =  ( a  e.  RR* ,  b  e.  RR*  |->  { x  e.  RR*  |  ( a  <_  x  /\  x  <  b ) } )
2 df-icc 11299 . 2  |-  [,]  =  ( a  e.  RR* ,  b  e.  RR*  |->  { x  e.  RR*  |  ( a  <_  x  /\  x  <_  b ) } )
3 idd 24 . 2  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <_  w  ->  A  <_  w ) )
4 xrltle 11118 . 2  |-  ( ( w  e.  RR*  /\  B  e.  RR* )  ->  (
w  <  B  ->  w  <_  B ) )
51, 2, 3, 4ixxssixx 11306 1  |-  ( A [,) B )  C_  ( A [,] B )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    e. wcel 1756    C_ wss 3323   class class class wbr 4287  (class class class)co 6086   RR*cxr 9409    < clt 9410    <_ cle 9411   [,)cico 11294   [,]cicc 11295
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 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-pre-lttri 9348  ax-pre-lttrn 9349
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 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-po 4636  df-so 4637  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-ico 11298  df-icc 11299
This theorem is referenced by:  eliccelico  26018  xrge0slmod  26264  xrge0iifcnv  26315  lmlimxrge0  26330  lmdvglim  26336  esumfsupre  26472  esumpfinvallem  26475  esumpfinval  26476  esumpfinvalf  26477  esumpcvgval  26479  esumpmono  26480  esummulc1  26482
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