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Theorem elicc1 11585
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 11548 . 2  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
21elixx1 11550 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 184    /\ wa 369    /\ w3a 973    e. wcel 1767   class class class wbr 4453  (class class class)co 6295   RR*cxr 9639    <_ cle 9641   [,]cicc 11544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-iota 5557  df-fun 5596  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-xr 9644  df-icc 11548
This theorem is referenced by:  iccid  11586  iccleub  11592  iccgelb  11593  elicc2  11601  elicc4  11603  elxrge0  11641  lbicc2  11648  ubicc2  11649  difreicc  11664  cnblcld  21150  oprpiece1res1  21319  ovolf  21761  volivth  21884  itg2ge0  22010  itg2const2  22016  taylfvallem1  22619  tayl0  22624  radcnvcl  22679  radcnvle  22682  psercnlem1  22687  eliccelico  27411  xrdifh  27414  xrge0neqmnf  27503  unitssxrge0  27707  esumle  27890  esumlef  27895  esumpinfsum  27908  voliune  28026  volfiniune  28027  ddemeas  28033  prob01  28177  ftc1cnnclem  30015  ftc1anc  30025  ftc2nc  30026  elicc3  30062  iocinico  31108  icoiccdif  31451  iblsplit  31607  iblspltprt  31614  itgspltprt  31620  fourierdlem1  31731  fourierdlem27  31757
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