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Theorem iooneg 9739
Description: Membership in a negated open real interval. (Contributed by Paul Chapman, 26-Nov-2007.)
Assertion
Ref Expression
iooneg  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  e.  ( A (,) B )  <->  -u C  e.  ( -u B (,) -u A ) ) )

Proof of Theorem iooneg
StepHypRef Expression
1 ltneg 8192 . . . . 5  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  <  C  <->  -u C  <  -u A
) )
213adant2 985 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  C  <->  -u C  <  -u A ) )
3 ltneg 8192 . . . . . 6  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  <  B  <->  -u B  <  -u C
) )
43ancoms 266 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( C  <  B  <->  -u B  <  -u C
) )
543adant1 984 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  <  B  <->  -u B  <  -u C ) )
62, 5anbi12d 464 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <  C  /\  C  <  B )  <-> 
( -u C  <  -u A  /\  -u B  <  -u C
) ) )
7 ancom 264 . . 3  |-  ( (
-u C  <  -u A  /\  -u B  <  -u C
)  <->  ( -u B  <  -u C  /\  -u C  <  -u A ) )
86, 7syl6bb 195 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <  C  /\  C  <  B )  <-> 
( -u B  <  -u C  /\  -u C  <  -u A
) ) )
9 rexr 7779 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
10 rexr 7779 . . 3  |-  ( B  e.  RR  ->  B  e.  RR* )
11 rexr 7779 . . 3  |-  ( C  e.  RR  ->  C  e.  RR* )
12 elioo5 9684 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( C  e.  ( A (,) B )  <->  ( A  <  C  /\  C  < 
B ) ) )
139, 10, 11, 12syl3an 1243 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  e.  ( A (,) B )  <->  ( A  <  C  /\  C  < 
B ) ) )
14 renegcl 7991 . . . 4  |-  ( B  e.  RR  ->  -u B  e.  RR )
15 renegcl 7991 . . . 4  |-  ( A  e.  RR  ->  -u A  e.  RR )
16 renegcl 7991 . . . 4  |-  ( C  e.  RR  ->  -u C  e.  RR )
17 rexr 7779 . . . . 5  |-  ( -u B  e.  RR  ->  -u B  e.  RR* )
18 rexr 7779 . . . . 5  |-  ( -u A  e.  RR  ->  -u A  e.  RR* )
19 rexr 7779 . . . . 5  |-  ( -u C  e.  RR  ->  -u C  e.  RR* )
20 elioo5 9684 . . . . 5  |-  ( (
-u B  e.  RR*  /\  -u A  e.  RR*  /\  -u C  e.  RR* )  ->  ( -u C  e.  ( -u B (,) -u A )  <->  ( -u B  <  -u C  /\  -u C  <  -u A ) ) )
2117, 18, 19, 20syl3an 1243 . . . 4  |-  ( (
-u B  e.  RR  /\  -u A  e.  RR  /\  -u C  e.  RR )  ->  ( -u C  e.  ( -u B (,) -u A )  <->  ( -u B  <  -u C  /\  -u C  <  -u A ) ) )
2214, 15, 16, 21syl3an 1243 . . 3  |-  ( ( B  e.  RR  /\  A  e.  RR  /\  C  e.  RR )  ->  ( -u C  e.  ( -u B (,) -u A )  <->  ( -u B  <  -u C  /\  -u C  <  -u A ) ) )
23223com12 1170 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( -u C  e.  ( -u B (,) -u A )  <->  ( -u B  <  -u C  /\  -u C  <  -u A ) ) )
248, 13, 233bitr4d 219 1  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  e.  ( A (,) B )  <->  -u C  e.  ( -u B (,) -u A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 947    e. wcel 1465   class class class wbr 3899  (class class class)co 5742   RRcr 7587   RR*cxr 7767    < clt 7768   -ucneg 7902   (,)cioo 9639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-addcom 7688  ax-addass 7690  ax-distr 7692  ax-i2m1 7693  ax-0id 7696  ax-rnegex 7697  ax-cnre 7699  ax-pre-ltadd 7704
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-iota 5058  df-fun 5095  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-sub 7903  df-neg 7904  df-ioo 9643
This theorem is referenced by: (None)
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