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Theorem iccneg 8857
Description: Membership in a negated closed real interval. (Contributed by Paul Chapman, 26-Nov-2007.)
Assertion
Ref Expression
iccneg  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  e.  ( A [,] B )  <->  -u C  e.  ( -u B [,] -u A ) ) )

Proof of Theorem iccneg
StepHypRef Expression
1 renegcl 7272 . . . . 5  |-  ( C  e.  RR  ->  -u C  e.  RR )
2 ax-1 5 . . . . 5  |-  ( C  e.  RR  ->  ( -u C  e.  RR  ->  C  e.  RR ) )
31, 2impbid2 131 . . . 4  |-  ( C  e.  RR  ->  ( C  e.  RR  <->  -u C  e.  RR ) )
433ad2ant3 927 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  e.  RR  <->  -u C  e.  RR ) )
5 ancom 253 . . . 4  |-  ( ( C  <_  B  /\  A  <_  C )  <->  ( A  <_  C  /\  C  <_  B ) )
6 leneg 7460 . . . . . . 7  |-  ( ( C  e.  RR  /\  B  e.  RR )  ->  ( C  <_  B  <->  -u B  <_  -u C ) )
76ancoms 255 . . . . . 6  |-  ( ( B  e.  RR  /\  C  e.  RR )  ->  ( C  <_  B  <->  -u B  <_  -u C ) )
873adant1 922 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  <_  B  <->  -u B  <_  -u C ) )
9 leneg 7460 . . . . . 6  |-  ( ( A  e.  RR  /\  C  e.  RR )  ->  ( A  <_  C  <->  -u C  <_  -u A ) )
1093adant2 923 . . . . 5  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <_  C  <->  -u C  <_  -u A ) )
118, 10anbi12d 442 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  <_  B  /\  A  <_  C )  <-> 
( -u B  <_  -u C  /\  -u C  <_  -u A
) ) )
125, 11syl5bbr 183 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( A  <_  C  /\  C  <_  B )  <-> 
( -u B  <_  -u C  /\  -u C  <_  -u A
) ) )
134, 12anbi12d 442 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  (
( C  e.  RR  /\  ( A  <_  C  /\  C  <_  B ) )  <->  ( -u C  e.  RR  /\  ( -u B  <_  -u C  /\  -u C  <_ 
-u A ) ) ) )
14 elicc2 8807 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( C  e.  ( A [,] B )  <-> 
( C  e.  RR  /\  A  <_  C  /\  C  <_  B ) ) )
15143adant3 924 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR  /\  A  <_  C  /\  C  <_  B
) ) )
16 3anass 889 . . 3  |-  ( ( C  e.  RR  /\  A  <_  C  /\  C  <_  B )  <->  ( C  e.  RR  /\  ( A  <_  C  /\  C  <_  B ) ) )
1715, 16syl6bb 185 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  RR  /\  ( A  <_  C  /\  C  <_  B ) ) ) )
18 renegcl 7272 . . . . 5  |-  ( B  e.  RR  ->  -u B  e.  RR )
19 renegcl 7272 . . . . 5  |-  ( A  e.  RR  ->  -u A  e.  RR )
20 elicc2 8807 . . . . 5  |-  ( (
-u B  e.  RR  /\  -u A  e.  RR )  ->  ( -u C  e.  ( -u B [,] -u A )  <->  ( -u C  e.  RR  /\  -u B  <_ 
-u C  /\  -u C  <_ 
-u A ) ) )
2118, 19, 20syl2anr 274 . . . 4  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( -u C  e.  ( -u B [,] -u A )  <->  ( -u C  e.  RR  /\  -u B  <_ 
-u C  /\  -u C  <_ 
-u A ) ) )
22213adant3 924 . . 3  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( -u C  e.  ( -u B [,] -u A )  <->  ( -u C  e.  RR  /\  -u B  <_ 
-u C  /\  -u C  <_ 
-u A ) ) )
23 3anass 889 . . 3  |-  ( (
-u C  e.  RR  /\  -u B  <_  -u C  /\  -u C  <_  -u A
)  <->  ( -u C  e.  RR  /\  ( -u B  <_  -u C  /\  -u C  <_ 
-u A ) ) )
2422, 23syl6bb 185 . 2  |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( -u C  e.  ( -u B [,] -u A )  <->  ( -u C  e.  RR  /\  ( -u B  <_  -u C  /\  -u C  <_ 
-u A ) ) ) )
2513, 17, 243bitr4d 209 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 97    <-> wb 98    /\ w3a 885    e. wcel 1393   class class class wbr 3764  (class class class)co 5512   RRcr 6888    <_ cle 7061   -ucneg 7183   [,]cicc 8760
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-1cn 6977  ax-1re 6978  ax-icn 6979  ax-addcl 6980  ax-addrcl 6981  ax-mulcl 6982  ax-addcom 6984  ax-addass 6986  ax-distr 6988  ax-i2m1 6989  ax-0id 6992  ax-rnegex 6993  ax-cnre 6995  ax-pre-ltirr 6996  ax-pre-ltwlin 6997  ax-pre-lttrn 6998  ax-pre-ltadd 7000
This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-id 4030  df-po 4033  df-iso 4034  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-iota 4867  df-fun 4904  df-fv 4910  df-riota 5468  df-ov 5515  df-oprab 5516  df-mpt2 5517  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065  df-le 7066  df-sub 7184  df-neg 7185  df-icc 8764
This theorem is referenced by: (None)
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