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Theorem iccgelb 9968
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 9960 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B
) ) )
21biimpa 296 . . 3  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,] B ) )  ->  ( C  e. 
RR*  /\  A  <_  C  /\  C  <_  B
) )
32simp2d 1012 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  ( A [,] B ) )  ->  A  <_  C
)
433impa 1196 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A [,] B
) )  ->  A  <_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    /\ w3a 980    e. wcel 2160   class class class wbr 4021  (class class class)co 5900   RR*cxr 8026    <_ cle 8028   [,]cicc 9927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-cnex 7937  ax-resscn 7938
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-iota 5199  df-fun 5240  df-fv 5246  df-ov 5903  df-oprab 5904  df-mpo 5905  df-pnf 8029  df-mnf 8030  df-xr 8031  df-icc 9931
This theorem is referenced by:  cos12dec  11816  suplociccreex  14587  suplociccex  14588  dedekindicc  14596
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