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Theorem elioo5 11573
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 11560 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A (,) B )  <->  ( C  e.  RR*  /\  A  < 
C  /\  C  <  B ) ) )
213adant3 1011 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( C  e.  ( A (,) B )  <->  ( C  e.  RR*  /\  A  < 
C  /\  C  <  B ) ) )
3 3anass 972 . . . 4  |-  ( ( C  e.  RR*  /\  A  <  C  /\  C  < 
B )  <->  ( C  e.  RR*  /\  ( A  <  C  /\  C  <  B ) ) )
43baibr 899 . . 3  |-  ( C  e.  RR*  ->  ( ( A  <  C  /\  C  <  B )  <->  ( C  e.  RR*  /\  A  < 
C  /\  C  <  B ) ) )
543ad2ant3 1014 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  (
( A  <  C  /\  C  <  B )  <-> 
( C  e.  RR*  /\  A  <  C  /\  C  <  B ) ) )
62, 5bitr4d 256 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 184    /\ wa 369    /\ w3a 968    e. wcel 1762   class class class wbr 4442  (class class class)co 6277   RR*cxr 9618    < clt 9619   (,)cioo 11520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-id 4790  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-iota 5544  df-fun 5583  df-fv 5589  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-xr 9623  df-ioo 11524
This theorem is referenced by:  iooshf  11594  iooneg  11631  lhop1  22145  tan2h  29613  ftc1anclem1  29656
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