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Theorem elioo5 11700
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 11684 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A (,) B )  <->  ( C  e.  RR*  /\  A  < 
C  /\  C  <  B ) ) )
213adant3 1025 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( C  e.  ( A (,) B )  <->  ( C  e.  RR*  /\  A  < 
C  /\  C  <  B ) ) )
3 3anass 986 . . . 4  |-  ( ( C  e.  RR*  /\  A  <  C  /\  C  < 
B )  <->  ( C  e.  RR*  /\  ( A  <  C  /\  C  <  B ) ) )
43baibr 912 . . 3  |-  ( C  e.  RR*  ->  ( ( A  <  C  /\  C  <  B )  <->  ( C  e.  RR*  /\  A  < 
C  /\  C  <  B ) ) )
543ad2ant3 1028 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A  <  C  /\  C  <  B )  <-> 
( C  e.  RR*  /\  A  <  C  /\  C  <  B ) ) )
62, 5bitr4d 259 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( C  e.  ( A (,) B )  <->  ( A  <  C  /\  C  < 
B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    e. wcel 1872   class class class wbr 4423  (class class class)co 6306   RR*cxr 9682    < clt 9683   (,)cioo 11643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660  ax-un 6598  ax-cnex 9603  ax-resscn 9604
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-xr 9687  df-ioo 11647
This theorem is referenced by:  iooshf  11721  iooneg  11760  lhop1  22965  tan2h  31902  poimir  31938  ftc1anclem1  31982
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