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Theorem iccleub 9742
Description: An element of a closed interval is less than or equal to its upper bound. (Contributed by Jeff Hankins, 14-Jul-2009.)
Assertion
Ref Expression
iccleub  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A [,] B
) )  ->  C  <_  B )

Proof of Theorem iccleub
StepHypRef Expression
1 elicc1 9735 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B
) ) )
2 simp3 984 . . 3  |-  ( ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B )  ->  C  <_  B )
31, 2syl6bi 162 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  ->  C  <_  B ) )
433impia 1179 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A [,] B
) )  ->  C  <_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 963    e. wcel 1481   class class class wbr 3935  (class class class)co 5780   RR*cxr 7821    <_ cle 7823   [,]cicc 9702
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4052  ax-pow 4104  ax-pr 4137  ax-un 4361  ax-setind 4458  ax-cnex 7733  ax-resscn 7734
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-dif 3076  df-un 3078  df-in 3080  df-ss 3087  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-br 3936  df-opab 3996  df-id 4221  df-xp 4551  df-rel 4552  df-cnv 4553  df-co 4554  df-dm 4555  df-iota 5094  df-fun 5131  df-fv 5137  df-ov 5783  df-oprab 5784  df-mpo 5785  df-pnf 7824  df-mnf 7825  df-xr 7826  df-icc 9706
This theorem is referenced by:  cos12dec  11503  suplociccreex  12803  suplociccex  12804  dedekindicc  12812
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