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Theorem iooneg 9711
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 8188 . . . . 5  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  <  C  <->  -u C  <  -u A
) )
213adant2 983 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  C  <->  -u C  <  -u A ) )
3 ltneg 8188 . . . . . 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 982 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  <  B  <->  -u B  <  -u C ) )
62, 5anbi12d 462 . . 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 7775 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
10 rexr 7775 . . 3  |-  ( B  e.  RR  ->  B  e.  RR* )
11 rexr 7775 . . 3  |-  ( C  e.  RR  ->  C  e.  RR* )
12 elioo5 9656 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( C  e.  ( A (,) B )  <->  ( A  <  C  /\  C  < 
B ) ) )
139, 10, 11, 12syl3an 1241 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  e.  ( A (,) B )  <->  ( A  <  C  /\  C  < 
B ) ) )
14 renegcl 7987 . . . 4  |-  ( B  e.  RR  ->  -u B  e.  RR )
15 renegcl 7987 . . . 4  |-  ( A  e.  RR  ->  -u A  e.  RR )
16 renegcl 7987 . . . 4  |-  ( C  e.  RR  ->  -u C  e.  RR )
17 rexr 7775 . . . . 5  |-  ( -u B  e.  RR  ->  -u B  e.  RR* )
18 rexr 7775 . . . . 5  |-  ( -u A  e.  RR  ->  -u A  e.  RR* )
19 rexr 7775 . . . . 5  |-  ( -u C  e.  RR  ->  -u C  e.  RR* )
20 elioo5 9656 . . . . 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 1241 . . . 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 1241 . . 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 1168 . 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 945    e. wcel 1463   class class class wbr 3897  (class class class)co 5740   RRcr 7583   RR*cxr 7763    < clt 7764   -ucneg 7898   (,)cioo 9611
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-addass 7686  ax-distr 7688  ax-i2m1 7689  ax-0id 7692  ax-rnegex 7693  ax-cnre 7695  ax-pre-ltadd 7700
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-iota 5056  df-fun 5093  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-sub 7899  df-neg 7900  df-ioo 9615
This theorem is referenced by: (None)
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