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Theorem elioo5 10800
Description: Membership in an open interval of extended reals. (Contributed by NM, 17-Aug-2008.)
Assertion
Ref Expression
elioo5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( C  e.  ( A (,) B )  <->  ( A  <  C  /\  C  < 
B ) ) )

Proof of Theorem elioo5
StepHypRef Expression
1 elioo1 10788 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A (,) B )  <->  ( C  e.  RR*  /\  A  < 
C  /\  C  <  B ) ) )
213adant3 975 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( C  e.  ( A (,) B )  <->  ( C  e.  RR*  /\  A  < 
C  /\  C  <  B ) ) )
3 3anass 938 . . . 4  |-  ( ( C  e.  RR*  /\  A  <  C  /\  C  < 
B )  <->  ( C  e.  RR*  /\  ( A  <  C  /\  C  <  B ) ) )
43baibr 872 . . 3  |-  ( C  e.  RR*  ->  ( ( A  <  C  /\  C  <  B )  <->  ( C  e.  RR*  /\  A  < 
C  /\  C  <  B ) ) )
543ad2ant3 978 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A  <  C  /\  C  <  B )  <-> 
( C  e.  RR*  /\  A  <  C  /\  C  <  B ) ) )
62, 5bitr4d 247 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 176    /\ wa 358    /\ w3a 934    e. wcel 1710   class class class wbr 4104  (class class class)co 5945   RR*cxr 8956    < clt 8957   (,)cioo 10748
This theorem is referenced by:  iooshf  10820  iooneg  10848  lhop1  19465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4222  ax-nul 4230  ax-pr 4295  ax-un 4594  ax-cnex 8883  ax-resscn 8884
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-opab 4159  df-id 4391  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-iota 5301  df-fun 5339  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-xr 8961  df-ioo 10752
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