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Theorem elicc1 11708
Description: Membership in a closed interval of extended reals. (Contributed by NM, 24-Dec-2006.) (Revised by Mario Carneiro, 3-Nov-2013.)
Assertion
Ref Expression
elicc1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B
) ) )

Proof of Theorem elicc1
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-icc 11670 . 2  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
21elixx1 11672 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ wa 375    /\ w3a 991    e. wcel 1897   class class class wbr 4415  (class class class)co 6314   RR*cxr 9699    <_ cle 9701   [,]cicc 11666
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652  ax-un 6609  ax-cnex 9620  ax-resscn 9621
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-uni 4212  df-br 4416  df-opab 4475  df-id 4767  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-iota 5564  df-fun 5602  df-fv 5608  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-xr 9704  df-icc 11670
This theorem is referenced by:  iccid  11709  iccleub  11718  iccgelb  11719  elicc2  11727  elicc4  11729  xrge0neqmnf  11765  elxrge0  11769  lbicc2  11776  ubicc2  11777  difreicc  11792  cnblcld  21843  oprpiece1res1  22027  ovolf  22483  volivth  22613  itg2ge0  22741  itg2const2  22747  taylfvallem1  23360  tayl0  23365  radcnvcl  23420  radcnvle  23423  psercnlem1  23428  eliccelico  28407  xrdifh  28410  unitssxrge0  28754  esumle  28927  esumlef  28931  esumpinfsum  28946  voliune  29100  volfiniune  29101  ddemeas  29107  prob01  29294  elicc3  31021  ftc1cnnclem  32059  ftc1anc  32069  ftc2nc  32070  iocinico  36140  icoiccdif  37662  iblsplit  37880  iblspltprt  37887  itgspltprt  37893  fourierdlem1  38007  iccpartrn  38781
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