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Theorem elicc1 10853
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 10816 . 2  |-  [,]  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <_  y ) } )
21elixx1 10818 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B
) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 935    e. wcel 1715   class class class wbr 4125  (class class class)co 5981   RR*cxr 9013    <_ cle 9015   [,]cicc 10812
This theorem is referenced by:  iccid  10854  iccleub  10860  elicc2  10868  elicc4  10870  elxrge0  10900  lbicc2  10905  ubicc2  10906  difreicc  10920  cnblcld  18497  oprpiece1res1  18664  ovolf  19056  volivth  19177  itg2ge0  19305  itg2const2  19311  taylfvallem1  19951  tayl0  19956  radcnvcl  20011  radcnvle  20014  psercnlem1  20019  iccgelb  23537  eliccelico  23541  xrdifh  23544  xrge0neqmnf  23603  unitssxrge0  23653  esumle  23914  esumlef  23919  esumpinfsum  23932  voliune  24048  volfiniune  24049  prob01  24119  itg2addnclem  25675  ftc1cnnclem  25696  elicc3  25735
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316  ax-un 4615  ax-cnex 8940  ax-resscn 8941
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-rab 2637  df-v 2875  df-sbc 3078  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-br 4126  df-opab 4180  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-iota 5322  df-fun 5360  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-xr 9018  df-icc 10816
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