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Theorem iooneg 8626
Description: Membership in a negated open real interval. (Contributed by Paul Chapman, 26-Nov-2007.)
Assertion
Ref Expression
iooneg  RR  RR  C  RR  C  (,)  -u C  -u (,) -u

Proof of Theorem iooneg
StepHypRef Expression
1 ltneg 7252 . . . . 5  RR  C  RR  <  C  -u C  <  -u
213adant2 922 . . . 4  RR  RR  C  RR  <  C  -u C  <  -u
3 ltneg 7252 . . . . . 6  C  RR  RR  C  <  -u  <  -u C
43ancoms 255 . . . . 5  RR  C  RR  C  <  -u  <  -u C
543adant1 921 . . . 4  RR  RR  C  RR  C  <  -u  <  -u C
62, 5anbi12d 442 . . 3  RR  RR  C  RR  <  C  C  <  -u C  <  -u  -u  <  -u C
7 ancom 253 . . 3 
-u C  <  -u  -u  <  -u C  -u  <  -u C  -u C  <  -u
86, 7syl6bb 185 . 2  RR  RR  C  RR  <  C  C  <  -u  <  -u C  -u C  <  -u
9 rexr 6868 . . 3  RR  RR*
10 rexr 6868 . . 3  RR  RR*
11 rexr 6868 . . 3  C  RR  C  RR*
12 elioo5 8572 . . 3  RR*  RR*  C 
RR*  C  (,)  <  C  C  <
139, 10, 11, 12syl3an 1176 . 2  RR  RR  C  RR  C  (,)  <  C  C  <
14 renegcl 7068 . . . 4  RR  -u  RR
15 renegcl 7068 . . . 4  RR  -u  RR
16 renegcl 7068 . . . 4  C  RR  -u C  RR
17 rexr 6868 . . . . 5  -u  RR  -u  RR*
18 rexr 6868 . . . . 5  -u  RR  -u  RR*
19 rexr 6868 . . . . 5  -u C  RR  -u C  RR*
20 elioo5 8572 . . . . 5 
-u  RR*  -u  RR*  -u C  RR*  -u C  -u (,) -u  -u  <  -u C  -u C  <  -u
2117, 18, 19, 20syl3an 1176 . . . 4 
-u  RR  -u  RR  -u C  RR  -u C  -u (,) -u  -u  <  -u C  -u C  <  -u
2214, 15, 16, 21syl3an 1176 . . 3  RR  RR  C  RR  -u C  -u (,) -u  -u  <  -u C  -u C  <  -u
23223com12 1107 . 2  RR  RR  C  RR  -u C  -u (,) -u  -u  <  -u C  -u C  <  -u
248, 13, 233bitr4d 209 1  RR  RR  C  RR  C  (,)  -u C  -u (,) -u
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wb 98   w3a 884   wcel 1390   class class class wbr 3755  (class class class)co 5455   RRcr 6710   RR*cxr 6856    < clt 6857   -ucneg 6980   (,)cioo 8527
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136  ax-setind 4220  ax-cnex 6774  ax-resscn 6775  ax-1cn 6776  ax-1re 6777  ax-icn 6778  ax-addcl 6779  ax-addrcl 6780  ax-mulcl 6781  ax-addcom 6783  ax-addass 6785  ax-distr 6787  ax-i2m1 6788  ax-0id 6791  ax-rnegex 6792  ax-cnre 6794  ax-pre-ltadd 6799
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-nel 2204  df-ral 2305  df-rex 2306  df-reu 2307  df-rab 2309  df-v 2553  df-sbc 2759  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-iota 4810  df-fun 4847  df-fv 4853  df-riota 5411  df-ov 5458  df-oprab 5459  df-mpt2 5460  df-pnf 6859  df-mnf 6860  df-xr 6861  df-ltxr 6862  df-sub 6981  df-neg 6982  df-ioo 8531
This theorem is referenced by: (None)
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