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Theorem iooneg 9075
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 7622 . . . . 5  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  <  C  <->  -u C  <  -u A
) )
213adant2 958 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  C  <->  -u C  <  -u A ) )
3 ltneg 7622 . . . . . 6  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  <  B  <->  -u B  <  -u C
) )
43ancoms 264 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( C  <  B  <->  -u B  <  -u C
) )
543adant1 957 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  <  B  <->  -u B  <  -u C ) )
62, 5anbi12d 457 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <  C  /\  C  <  B )  <-> 
( -u C  <  -u A  /\  -u B  <  -u C
) ) )
7 ancom 262 . . 3  |-  ( (
-u C  <  -u A  /\  -u B  <  -u C
)  <->  ( -u B  <  -u C  /\  -u C  <  -u A ) )
86, 7syl6bb 194 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <  C  /\  C  <  B )  <-> 
( -u B  <  -u C  /\  -u C  <  -u A
) ) )
9 rexr 7215 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
10 rexr 7215 . . 3  |-  ( B  e.  RR  ->  B  e.  RR* )
11 rexr 7215 . . 3  |-  ( C  e.  RR  ->  C  e.  RR* )
12 elioo5 9021 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( C  e.  ( A (,) B )  <->  ( A  <  C  /\  C  < 
B ) ) )
139, 10, 11, 12syl3an 1212 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  e.  ( A (,) B )  <->  ( A  <  C  /\  C  < 
B ) ) )
14 renegcl 7425 . . . 4  |-  ( B  e.  RR  ->  -u B  e.  RR )
15 renegcl 7425 . . . 4  |-  ( A  e.  RR  ->  -u A  e.  RR )
16 renegcl 7425 . . . 4  |-  ( C  e.  RR  ->  -u C  e.  RR )
17 rexr 7215 . . . . 5  |-  ( -u B  e.  RR  ->  -u B  e.  RR* )
18 rexr 7215 . . . . 5  |-  ( -u A  e.  RR  ->  -u A  e.  RR* )
19 rexr 7215 . . . . 5  |-  ( -u C  e.  RR  ->  -u C  e.  RR* )
20 elioo5 9021 . . . . 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 1212 . . . 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 1212 . . 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 1143 . 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 218 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 102    <-> wb 103    /\ w3a 920    e. wcel 1434   class class class wbr 3787  (class class class)co 5537   RRcr 7031   RR*cxr 7203    < clt 7204   -ucneg 7336   (,)cioo 8976
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-cnex 7118  ax-resscn 7119  ax-1cn 7120  ax-1re 7121  ax-icn 7122  ax-addcl 7123  ax-addrcl 7124  ax-mulcl 7125  ax-addcom 7127  ax-addass 7129  ax-distr 7131  ax-i2m1 7132  ax-0id 7135  ax-rnegex 7136  ax-cnre 7138  ax-pre-ltadd 7143
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-id 4050  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-iota 4891  df-fun 4928  df-fv 4934  df-riota 5493  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-pnf 7206  df-mnf 7207  df-xr 7208  df-ltxr 7209  df-sub 7337  df-neg 7338  df-ioo 8980
This theorem is referenced by: (None)
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