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Theorem elicc4 10977
Description: Membership in a closed real interval. (Contributed by Stefan O'Rear, 16-Nov-2014.) (Proof shortened by Mario Carneiro, 1-Jan-2017.)
Assertion
Ref Expression
elicc4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( C  e.  ( A [,] B )  <->  ( A  <_  C  /\  C  <_  B ) ) )

Proof of Theorem elicc4
StepHypRef Expression
1 elicc1 10960 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B
) ) )
2 3anass 940 . . . 4  |-  ( ( C  e.  RR*  /\  A  <_  C  /\  C  <_  B )  <->  ( C  e.  RR*  /\  ( A  <_  C  /\  C  <_  B ) ) )
31, 2syl6bb 253 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR*  /\  ( A  <_  C  /\  C  <_  B ) ) ) )
43baibd 876 . 2  |-  ( ( ( A  e.  RR*  /\  B  e.  RR* )  /\  C  e.  RR* )  ->  ( C  e.  ( A [,] B )  <-> 
( A  <_  C  /\  C  <_  B ) ) )
543impa 1148 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( C  e.  ( A [,] B )  <->  ( A  <_  C  /\  C  <_  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1725   class class class wbr 4212  (class class class)co 6081   RR*cxr 9119    <_ cle 9121   [,]cicc 10919
This theorem is referenced by:  elicc4abs  12123  xrge0addass  24211  esumle  24449  esumlef  24454
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-iota 5418  df-fun 5456  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-xr 9124  df-icc 10923
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