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Theorem elioo2 9402
Description: Membership in an open interval of extended reals. (Contributed by NM, 6-Feb-2007.)
Assertion
Ref Expression
elioo2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A (,) B )  <->  ( C  e.  RR  /\  A  < 
C  /\  C  <  B ) ) )

Proof of Theorem elioo2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 iooval2 9396 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A (,) B )  =  { x  e.  RR  |  ( A  < 
x  /\  x  <  B ) } )
21eleq2d 2158 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A (,) B )  <->  C  e.  { x  e.  RR  | 
( A  <  x  /\  x  <  B ) } ) )
3 breq2 3857 . . . . 5  |-  ( x  =  C  ->  ( A  <  x  <->  A  <  C ) )
4 breq1 3856 . . . . 5  |-  ( x  =  C  ->  (
x  <  B  <->  C  <  B ) )
53, 4anbi12d 458 . . . 4  |-  ( x  =  C  ->  (
( A  <  x  /\  x  <  B )  <-> 
( A  <  C  /\  C  <  B ) ) )
65elrab 2774 . . 3  |-  ( C  e.  { x  e.  RR  |  ( A  <  x  /\  x  <  B ) }  <->  ( C  e.  RR  /\  ( A  <  C  /\  C  <  B ) ) )
7 3anass 929 . . 3  |-  ( ( C  e.  RR  /\  A  <  C  /\  C  <  B )  <->  ( C  e.  RR  /\  ( A  <  C  /\  C  <  B ) ) )
86, 7bitr4i 186 . 2  |-  ( C  e.  { x  e.  RR  |  ( A  <  x  /\  x  <  B ) }  <->  ( C  e.  RR  /\  A  < 
C  /\  C  <  B ) )
92, 8syl6bb 195 1  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  ( C  e.  ( A (,) B )  <->  ( C  e.  RR  /\  A  < 
C  /\  C  <  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 925    = wceq 1290    e. wcel 1439   {crab 2364   class class class wbr 3853  (class class class)co 5668   RRcr 7412   RR*cxr 7584    < clt 7585   (,)cioo 9369
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3965  ax-pow 4017  ax-pr 4047  ax-un 4271  ax-setind 4368  ax-cnex 7499  ax-resscn 7500  ax-pre-ltirr 7520  ax-pre-ltwlin 7521  ax-pre-lttrn 7522
This theorem depends on definitions:  df-bi 116  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-nel 2352  df-ral 2365  df-rex 2366  df-rab 2369  df-v 2624  df-sbc 2844  df-dif 3004  df-un 3006  df-in 3008  df-ss 3015  df-pw 3437  df-sn 3458  df-pr 3459  df-op 3461  df-uni 3662  df-br 3854  df-opab 3908  df-id 4131  df-po 4134  df-iso 4135  df-xp 4460  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-iota 4995  df-fun 5032  df-fv 5038  df-ov 5671  df-oprab 5672  df-mpt2 5673  df-pnf 7587  df-mnf 7588  df-xr 7589  df-ltxr 7590  df-le 7591  df-ioo 9373
This theorem is referenced by:  eliooord  9409  elioopnf  9448  elioomnf  9449
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