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Theorem elioo5 11593
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 11580 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A (,) B )  <->  ( C  e.  RR*  /\  A  < 
C  /\  C  <  B ) ) )
213adant3 1017 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( C  e.  ( A (,) B )  <->  ( C  e.  RR*  /\  A  < 
C  /\  C  <  B ) ) )
3 3anass 978 . . . 4  |-  ( ( C  e.  RR*  /\  A  <  C  /\  C  < 
B )  <->  ( C  e.  RR*  /\  ( A  <  C  /\  C  <  B ) ) )
43baibr 904 . . 3  |-  ( C  e.  RR*  ->  ( ( A  <  C  /\  C  <  B )  <->  ( C  e.  RR*  /\  A  < 
C  /\  C  <  B ) ) )
543ad2ant3 1020 . 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 974    e. wcel 1804   class class class wbr 4437  (class class class)co 6281   RR*cxr 9630    < clt 9631   (,)cioo 11540
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3097  df-sbc 3314  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-iota 5541  df-fun 5580  df-fv 5586  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-xr 9635  df-ioo 11544
This theorem is referenced by:  iooshf  11614  iooneg  11651  lhop1  22393  tan2h  30023  ftc1anclem1  30066
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