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Theorem iooneg 11611
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 10013 . . . . 5  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  <  C  <->  -u C  <  -u A
) )
213adant2 1016 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <  C  <->  -u C  <  -u A ) )
3 ltneg 10013 . . . . . 6  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  <  B  <->  -u B  <  -u C
) )
43ancoms 451 . . . . 5  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( C  <  B  <->  -u B  <  -u C
) )
543adant1 1015 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  <  B  <->  -u B  <  -u C ) )
62, 5anbi12d 709 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <  C  /\  C  <  B )  <-> 
( -u C  <  -u A  /\  -u B  <  -u C
) ) )
7 ancom 448 . . 3  |-  ( (
-u C  <  -u A  /\  -u B  <  -u C
)  <->  ( -u B  <  -u C  /\  -u C  <  -u A ) )
86, 7syl6bb 261 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <  C  /\  C  <  B )  <-> 
( -u B  <  -u C  /\  -u C  <  -u A
) ) )
9 rexr 9589 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
10 rexr 9589 . . 3  |-  ( B  e.  RR  ->  B  e.  RR* )
11 rexr 9589 . . 3  |-  ( C  e.  RR  ->  C  e.  RR* )
12 elioo5 11553 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e. 
RR* )  ->  ( C  e.  ( A (,) B )  <->  ( A  <  C  /\  C  < 
B ) ) )
139, 10, 11, 12syl3an 1272 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  e.  ( A (,) B )  <->  ( A  <  C  /\  C  < 
B ) ) )
14 renegcl 9838 . . . 4  |-  ( B  e.  RR  ->  -u B  e.  RR )
15 renegcl 9838 . . . 4  |-  ( A  e.  RR  ->  -u A  e.  RR )
16 renegcl 9838 . . . 4  |-  ( C  e.  RR  ->  -u C  e.  RR )
17 rexr 9589 . . . . 5  |-  ( -u B  e.  RR  ->  -u B  e.  RR* )
18 rexr 9589 . . . . 5  |-  ( -u A  e.  RR  ->  -u A  e.  RR* )
19 rexr 9589 . . . . 5  |-  ( -u C  e.  RR  ->  -u C  e.  RR* )
20 elioo5 11553 . . . . 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 1272 . . . 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 1272 . . 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 1201 . 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 285 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    e. wcel 1842   class class class wbr 4394  (class class class)co 6234   RRcr 9441   RR*cxr 9577    < clt 9578   -ucneg 9762   (,)cioo 11500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6530  ax-cnex 9498  ax-resscn 9499  ax-1cn 9500  ax-icn 9501  ax-addcl 9502  ax-addrcl 9503  ax-mulcl 9504  ax-mulrcl 9505  ax-mulcom 9506  ax-addass 9507  ax-mulass 9508  ax-distr 9509  ax-i2m1 9510  ax-1ne0 9511  ax-1rid 9512  ax-rnegex 9513  ax-rrecex 9514  ax-cnre 9515  ax-pre-lttri 9516  ax-pre-lttrn 9517  ax-pre-ltadd 9518
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-po 4743  df-so 4744  df-xp 4948  df-rel 4949  df-cnv 4950  df-co 4951  df-dm 4952  df-rn 4953  df-res 4954  df-ima 4955  df-iota 5489  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6196  df-ov 6237  df-oprab 6238  df-mpt2 6239  df-er 7268  df-en 7475  df-dom 7476  df-sdom 7477  df-pnf 9580  df-mnf 9581  df-xr 9582  df-ltxr 9583  df-le 9584  df-sub 9763  df-neg 9764  df-ioo 11504
This theorem is referenced by:  lhop2  22600  asinsin  23440  atanlogsub  23464  atanbnd  23474
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