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Theorem iocssicc 11666
Description: A closed-above, open-below interval is a subset of its closure. (Contributed by Thierry Arnoux, 1-Apr-2017.)
Assertion
Ref Expression
iocssicc  |-  ( A (,] B )  C_  ( A [,] B )

Proof of Theorem iocssicc
Dummy variables  a 
b  w  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ioc 11587 . 2  |-  (,]  =  ( a  e.  RR* ,  b  e.  RR*  |->  { x  e.  RR*  |  ( a  <  x  /\  x  <_  b ) } )
2 df-icc 11589 . 2  |-  [,]  =  ( a  e.  RR* ,  b  e.  RR*  |->  { x  e.  RR*  |  ( a  <_  x  /\  x  <_  b ) } )
3 xrltle 11408 . 2  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  ->  A  <_  w ) )
4 idd 24 . 2  |-  ( ( w  e.  RR*  /\  B  e.  RR* )  ->  (
w  <_  B  ->  w  <_  B ) )
51, 2, 3, 4ixxssixx 11596 1  |-  ( A (,] B )  C_  ( A [,] B )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 367    e. wcel 1842    C_ wss 3414   class class class wbr 4395  (class class class)co 6278   RR*cxr 9657    < clt 9658    <_ cle 9659   (,]cioc 11583   [,]cicc 11585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-cnex 9578  ax-resscn 9579  ax-pre-lttri 9596  ax-pre-lttrn 9597
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-po 4744  df-so 4745  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-ioc 11587  df-icc 11589
This theorem is referenced by:  xrge0iifcnv  28368  xrge0iifcv  28369  xrge0iifhom  28372  pnfneige0  28386  lmxrge0  28387  eliccelioc  36929  limcicciooub  37011  fourierdlem17  37274  fourierdlem35  37292  fourierdlem41  37298  fourierdlem48  37305  fourierdlem49  37306  fourierdlem51  37308  fourierdlem71  37328  fourierdlem102  37359  fourierdlem114  37371
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