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Theorem iccssxr 11370
Description: A closed interval is a set of extended reals. (Contributed by FL, 28-Jul-2008.) (Revised by Mario Carneiro, 4-Jul-2014.)
Assertion
Ref Expression
iccssxr  |-  ( A [,] B )  C_  RR*

Proof of Theorem iccssxr
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-icc 11299 . 2  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
21ixxssxr 11304 1  |-  ( A [,] B )  C_  RR*
Colors of variables: wff setvar class
Syntax hints:    C_ wss 3323  (class class class)co 6086   RR*cxr 9409    <_ cle 9411   [,]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
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-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-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  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-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-1st 6572  df-2nd 6573  df-xr 9414  df-icc 11299
This theorem is referenced by:  supicclub2  11428  lecldbas  18798  ordtresticc  18802  prdsxmetlem  19918  xrge0gsumle  20385  xrge0tsms  20386  metdscn  20407  iccpnfhmeo  20492  xrhmeo  20493  volsup  21012  volsup2  21060  volivth  21062  itg2le  21192  itg2const2  21194  itg2lea  21197  itg2eqa  21198  itg2split  21202  itg2gt0  21213  dvgt0lem1  21449  radcnvlt1  21858  radcnvle  21860  pserulm  21862  psercnlem2  21864  psercnlem1  21865  psercn  21866  pserdvlem1  21867  pserdvlem2  21868  abelthlem3  21873  abelth  21881  logtayl  22080  xrge0infss  26004  xrge0infssd  26005  xrge0base  26097  xrge00  26098  xrge0mulgnn0  26101  xrge0addass  26102  xrge0nre  26104  xrge0addgt0  26105  xrge0adddir  26106  xrge0adddi  26107  xrge0npcan  26108  xrge0omnd  26125  xrge0tsmsd  26204  xrge0slmod  26264  xrge0iifiso  26317  xrge0iifhmeo  26318  xrge0pluscn  26322  xrge0mulc1cn  26323  xrge0tmdOLD  26327  lmlimxrge0  26330  pnfneige0  26333  lmxrge0  26334  esumle  26460  esummono  26461  gsumesum  26462  esumlub  26463  esumlef  26465  esumcst  26466  esumfsup  26471  esumpinfval  26474  esumpfinvallem  26475  esumpinfsum  26478  esumpmono  26480  esummulc2  26483  esumdivc  26484  hasheuni  26486  esumcvg  26487  measun  26577  measunl  26582  measiun  26584  voliune  26597  volfiniune  26598  ddemeas  26604  omsfval  26661  oms0  26662  probmeasb  26765  mblfinlem1  28381  itg2addnclem  28396  ftc1anc  28428
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