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Theorem iocssicc 11601
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 11523 . 2  |-  (,]  =  ( a  e.  RR* ,  b  e.  RR*  |->  { x  e.  RR*  |  ( a  <  x  /\  x  <_  b ) } )
2 df-icc 11525 . 2  |-  [,]  =  ( a  e.  RR* ,  b  e.  RR*  |->  { x  e.  RR*  |  ( a  <_  x  /\  x  <_  b ) } )
3 xrltle 11344 . 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 11532 1  |-  ( A (,] B )  C_  ( A [,] B )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    e. wcel 1762    C_ wss 3469   class class class wbr 4440  (class class class)co 6275   RR*cxr 9616    < clt 9617    <_ cle 9618   (,]cioc 11519   [,]cicc 11521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-pre-lttri 9555  ax-pre-lttrn 9556
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-ioc 11523  df-icc 11525
This theorem is referenced by:  xrge0iifcnv  27537  xrge0iifcv  27538  xrge0iifhom  27541  pnfneige0  27555  lmxrge0  27556  eliccelioc  31080  limcicciooub  31134  cncfiooicclem1  31187  fourierdlem17  31379  fourierdlem35  31397  fourierdlem41  31403  fourierdlem48  31410  fourierdlem49  31411  fourierdlem51  31413  fourierdlem71  31433  fourierdlem102  31464  fourierdlem114  31476
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