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Theorem iccgelb 11591
Description: An element of a closed interval is more than or equal to its lower bound (Contributed by Thierry Arnoux, 23-Dec-2016.)
Assertion
Ref Expression
iccgelb  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A [,] B
) )  ->  A  <_  C )

Proof of Theorem iccgelb
StepHypRef Expression
1 elicc1 11583 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B
) ) )
21biimpa 484 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,] B ) )  ->  ( C  e. 
RR*  /\  A  <_  C  /\  C  <_  B
) )
32simp2d 1009 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,] B ) )  ->  A  <_  C
)
433impa 1191 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A [,] B
) )  ->  A  <_  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 973    e. wcel 1767   class class class wbr 4452  (class class class)co 6294   RR*cxr 9637    <_ cle 9639   [,]cicc 11542
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 4573  ax-nul 4581  ax-pr 4691  ax-un 6586  ax-cnex 9558  ax-resscn 9559
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 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-opab 4511  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-iota 5556  df-fun 5595  df-fv 5601  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-xr 9642  df-icc 11546
This theorem is referenced by:  supicc  11678  ttgcontlem1  23979  xrge0infss  27371  xrge0addgt0  27473  xrge0adddir  27474  esumcst  27864  esumpinfval  27872  oms0  28059  probmeasb  28162  areaquad  31081  lefldiveq  31350  eliccelioc  31416  iccintsng  31418  cncfiooiccre  31525  itgioocnicc  31586  itgspltprt  31588  itgiccshift  31589  fourierdlem1  31699  fourierdlem6  31704  fourierdlem20  31718  fourierdlem24  31722  fourierdlem25  31723  fourierdlem27  31725  fourierdlem43  31741  fourierdlem44  31742  fourierdlem50  31748  fourierdlem51  31749  fourierdlem52  31750  fourierdlem56  31754  fourierdlem64  31762  fourierdlem73  31771  fourierdlem76  31774  fourierdlem81  31779  fourierdlem92  31790  fourierdlem102  31800  fourierdlem103  31801  fourierdlem104  31802  fourierdlem114  31812
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