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Theorem iccleub 10923
Description: An element of a closed interval is less than or equal to its upper bound. (Contributed by Jeffrey 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 10916 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B
) ) )
2 simp3 959 . . 3  |-  ( ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B )  ->  C  <_  B )
31, 2syl6bi 220 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  ->  C  <_  B ) )
433impia 1150 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A [,] B
) )  ->  C  <_  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    e. wcel 1721   class class class wbr 4172  (class class class)co 6040   RR*cxr 9075    <_ cle 9077   [,]cicc 10875
This theorem is referenced by:  oprpiece1res1  18929  ivthlem1  19301  isosctrlem1  20615  mblfinlem  26143  ftc1cnnclem  26177
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660  ax-cnex 9002  ax-resscn 9003
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-xr 9080  df-icc 10879
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