Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  icossicc Unicode version

Theorem icossicc 24112
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 10906 . 2  |-  [,)  =  ( a  e.  RR* ,  b  e.  RR*  |->  { x  e.  RR*  |  ( a  <_  x  /\  x  <  b ) } )
2 df-icc 10907 . 2  |-  [,]  =  ( a  e.  RR* ,  b  e.  RR*  |->  { x  e.  RR*  |  ( a  <_  x  /\  x  <_  b ) } )
3 idd 22 . 2  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <_  w  ->  A  <_  w ) )
4 xrltle 10726 . 2  |-  ( ( w  e.  RR*  /\  B  e.  RR* )  ->  (
w  <  B  ->  w  <_  B ) )
51, 2, 3, 4ixxssixx 10914 1  |-  ( A [,) B )  C_  ( A [,] B )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    e. wcel 1725    C_ wss 3307   class class class wbr 4199  (class class class)co 6067   RR*cxr 9103    < clt 9104    <_ cle 9105   [,)cico 10902   [,]cicc 10903
This theorem is referenced by:  eliccelico  24123  xrge0iifcnv  24302  lmlimxrge0  24317  lmdvglim  24322  esumfsupre  24444  esumpfinvallem  24447  esumpfinval  24448  esumpfinvalf  24449  esumpcvgval  24451  esumpmono  24452  esummulc1  24454  sitmcl  24646  itg2addnc  26200
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 2411  ax-sep 4317  ax-nul 4325  ax-pow 4364  ax-pr 4390  ax-un 4687  ax-cnex 9030  ax-resscn 9031  ax-pre-lttri 9048  ax-pre-lttrn 9049
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 2284  df-mo 2285  df-clab 2417  df-cleq 2423  df-clel 2426  df-nfc 2555  df-ne 2595  df-nel 2596  df-ral 2697  df-rex 2698  df-rab 2701  df-v 2945  df-sbc 3149  df-csb 3239  df-dif 3310  df-un 3312  df-in 3314  df-ss 3321  df-nul 3616  df-if 3727  df-pw 3788  df-sn 3807  df-pr 3808  df-op 3810  df-uni 4003  df-br 4200  df-opab 4254  df-mpt 4255  df-id 4485  df-po 4490  df-so 4491  df-xp 4870  df-rel 4871  df-cnv 4872  df-co 4873  df-dm 4874  df-rn 4875  df-res 4876  df-ima 4877  df-iota 5404  df-fun 5442  df-fn 5443  df-f 5444  df-f1 5445  df-fo 5446  df-f1o 5447  df-fv 5448  df-ov 6070  df-oprab 6071  df-mpt2 6072  df-er 6891  df-en 7096  df-dom 7097  df-sdom 7098  df-pnf 9106  df-mnf 9107  df-xr 9108  df-ltxr 9109  df-le 9110  df-ico 10906  df-icc 10907
  Copyright terms: Public domain W3C validator