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Theorem ixxub 11546
Description: Extract the upper bound of an interval. (Contributed by Mario Carneiro, 17-Jun-2014.)
Hypotheses
Ref Expression
ixx.1  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
ixxub.2  |-  ( ( w  e.  RR*  /\  B  e.  RR* )  ->  (
w  <  B  ->  w S B ) )
ixxub.3  |-  ( ( w  e.  RR*  /\  B  e.  RR* )  ->  (
w S B  ->  w  <_  B ) )
ixxub.4  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  ->  A R w ) )
ixxub.5  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A R w  ->  A  <_  w ) )
Assertion
Ref Expression
ixxub  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  sup (
( A O B ) ,  RR* ,  <  )  =  B )
Distinct variable groups:    x, w, y, z, A    w, O    w, B, x, y, z   
x, R, y, z   
x, S, y, z
Allowed substitution hints:    R( w)    S( w)    O( x, y, z)

Proof of Theorem ixxub
StepHypRef Expression
1 ixx.1 . . . . . . . . 9  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
21elixx1 11534 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A O B )  <->  ( w  e.  RR*  /\  A R w  /\  w S B ) ) )
323adant3 1016 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( w  e.  ( A O B )  <->  ( w  e.  RR*  /\  A R w  /\  w S B ) ) )
43biimpa 484 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  (
w  e.  RR*  /\  A R w  /\  w S B ) )
54simp3d 1010 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  w S B )
64simp1d 1008 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  w  e.  RR* )
7 simp2 997 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  B  e. 
RR* )
87adantr 465 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  B  e.  RR* )
9 ixxub.3 . . . . . 6  |-  ( ( w  e.  RR*  /\  B  e.  RR* )  ->  (
w S B  ->  w  <_  B ) )
106, 8, 9syl2anc 661 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  (
w S B  ->  w  <_  B ) )
115, 10mpd 15 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  w  <_  B )
1211ralrimiva 2878 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  A. w  e.  ( A O B ) w  <_  B
)
136ex 434 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( w  e.  ( A O B )  ->  w  e.  RR* ) )
1413ssrdv 3510 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( A O B )  C_  RR* )
15 supxrleub 11514 . . . 4  |-  ( ( ( A O B )  C_  RR*  /\  B  e.  RR* )  ->  ( sup ( ( A O B ) ,  RR* ,  <  )  <_  B  <->  A. w  e.  ( A O B ) w  <_  B ) )
1614, 7, 15syl2anc 661 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( sup ( ( A O B ) ,  RR* ,  <  )  <_  B  <->  A. w  e.  ( A O B ) w  <_  B ) )
1712, 16mpbird 232 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  sup (
( A O B ) ,  RR* ,  <  )  <_  B )
18 simprl 755 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  < 
B ) )  ->  sup ( ( A O B ) ,  RR* ,  <  )  <  w
)
1914ad2antrr 725 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  < 
B ) )  -> 
( A O B )  C_  RR* )
20 qre 11183 . . . . . . . . . . 11  |-  ( w  e.  QQ  ->  w  e.  RR )
2120rexrd 9639 . . . . . . . . . 10  |-  ( w  e.  QQ  ->  w  e.  RR* )
2221ad2antlr 726 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  < 
B ) )  ->  w  e.  RR* )
23 simp1 996 . . . . . . . . . . . 12  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  A  e. 
RR* )
2423ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  < 
B ) )  ->  A  e.  RR* )
25 supxrcl 11502 . . . . . . . . . . . . 13  |-  ( ( A O B ) 
C_  RR*  ->  sup (
( A O B ) ,  RR* ,  <  )  e.  RR* )
2614, 25syl 16 . . . . . . . . . . . 12  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  sup (
( A O B ) ,  RR* ,  <  )  e.  RR* )
2726ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  < 
B ) )  ->  sup ( ( A O B ) ,  RR* ,  <  )  e.  RR* )
28 simp3 998 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( A O B )  =/=  (/) )
29 n0 3794 . . . . . . . . . . . . . 14  |-  ( ( A O B )  =/=  (/)  <->  E. w  w  e.  ( A O B ) )
3028, 29sylib 196 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  E. w  w  e.  ( A O B ) )
3123adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  A  e.  RR* )
3226adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  sup ( ( A O B ) ,  RR* ,  <  )  e.  RR* )
334simp2d 1009 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  A R w )
34 ixxub.5 . . . . . . . . . . . . . . . 16  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A R w  ->  A  <_  w ) )
3531, 6, 34syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  ( A R w  ->  A  <_  w ) )
3633, 35mpd 15 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  A  <_  w )
37 supxrub 11512 . . . . . . . . . . . . . . 15  |-  ( ( ( A O B )  C_  RR*  /\  w  e.  ( A O B ) )  ->  w  <_  sup ( ( A O B ) , 
RR* ,  <  ) )
3814, 37sylan 471 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  w  <_  sup ( ( A O B ) , 
RR* ,  <  ) )
3931, 6, 32, 36, 38xrletrd 11361 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  A  <_  sup ( ( A O B ) , 
RR* ,  <  ) )
4030, 39exlimddv 1702 . . . . . . . . . . . 12  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  A  <_  sup ( ( A O B ) ,  RR* ,  <  ) )
4140ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  < 
B ) )  ->  A  <_  sup ( ( A O B ) , 
RR* ,  <  ) )
4224, 27, 22, 41, 18xrlelttrd 11359 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  < 
B ) )  ->  A  <  w )
43 ixxub.4 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  ->  A R w ) )
4424, 22, 43syl2anc 661 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  < 
B ) )  -> 
( A  <  w  ->  A R w ) )
4542, 44mpd 15 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  < 
B ) )  ->  A R w )
46 simprr 756 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  < 
B ) )  ->  w  <  B )
477ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  < 
B ) )  ->  B  e.  RR* )
48 ixxub.2 . . . . . . . . . . 11  |-  ( ( w  e.  RR*  /\  B  e.  RR* )  ->  (
w  <  B  ->  w S B ) )
4922, 47, 48syl2anc 661 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  < 
B ) )  -> 
( w  <  B  ->  w S B ) )
5046, 49mpd 15 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  < 
B ) )  ->  w S B )
513ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  < 
B ) )  -> 
( w  e.  ( A O B )  <-> 
( w  e.  RR*  /\  A R w  /\  w S B ) ) )
5222, 45, 50, 51mpbir3and 1179 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  < 
B ) )  ->  w  e.  ( A O B ) )
5319, 52, 37syl2anc 661 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  < 
B ) )  ->  w  <_  sup ( ( A O B ) , 
RR* ,  <  ) )
54 xrlenlt 9648 . . . . . . . 8  |-  ( ( w  e.  RR*  /\  sup ( ( A O B ) ,  RR* ,  <  )  e.  RR* )  ->  ( w  <_  sup ( ( A O B ) ,  RR* ,  <  )  <->  -.  sup (
( A O B ) ,  RR* ,  <  )  <  w ) )
5522, 27, 54syl2anc 661 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  < 
B ) )  -> 
( w  <_  sup ( ( A O B ) ,  RR* ,  <  )  <->  -.  sup (
( A O B ) ,  RR* ,  <  )  <  w ) )
5653, 55mpbid 210 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  < 
B ) )  ->  -.  sup ( ( A O B ) , 
RR* ,  <  )  < 
w )
5718, 56pm2.65da 576 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  ->  -.  ( sup ( ( A O B ) , 
RR* ,  <  )  < 
w  /\  w  <  B ) )
5857nrexdv 2920 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  -.  E. w  e.  QQ  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  <  B ) )
59 qbtwnxr 11395 . . . . . 6  |-  ( ( sup ( ( A O B ) , 
RR* ,  <  )  e. 
RR*  /\  B  e.  RR* 
/\  sup ( ( A O B ) , 
RR* ,  <  )  < 
B )  ->  E. w  e.  QQ  ( sup (
( A O B ) ,  RR* ,  <  )  <  w  /\  w  <  B ) )
60593expia 1198 . . . . 5  |-  ( ( sup ( ( A O B ) , 
RR* ,  <  )  e. 
RR*  /\  B  e.  RR* )  ->  ( sup ( ( A O B ) ,  RR* ,  <  )  <  B  ->  E. w  e.  QQ  ( sup ( ( A O B ) , 
RR* ,  <  )  < 
w  /\  w  <  B ) ) )
6126, 7, 60syl2anc 661 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( sup ( ( A O B ) ,  RR* ,  <  )  <  B  ->  E. w  e.  QQ  ( sup ( ( A O B ) , 
RR* ,  <  )  < 
w  /\  w  <  B ) ) )
6258, 61mtod 177 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  -.  sup ( ( A O B ) ,  RR* ,  <  )  <  B
)
63 xrlenlt 9648 . . . 4  |-  ( ( B  e.  RR*  /\  sup ( ( A O B ) ,  RR* ,  <  )  e.  RR* )  ->  ( B  <_  sup ( ( A O B ) ,  RR* ,  <  )  <->  -.  sup (
( A O B ) ,  RR* ,  <  )  <  B ) )
647, 26, 63syl2anc 661 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( B  <_  sup ( ( A O B ) , 
RR* ,  <  )  <->  -.  sup (
( A O B ) ,  RR* ,  <  )  <  B ) )
6562, 64mpbird 232 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  B  <_  sup ( ( A O B ) ,  RR* ,  <  ) )
66 xrletri3 11354 . . 3  |-  ( ( sup ( ( A O B ) , 
RR* ,  <  )  e. 
RR*  /\  B  e.  RR* )  ->  ( sup ( ( A O B ) ,  RR* ,  <  )  =  B  <-> 
( sup ( ( A O B ) ,  RR* ,  <  )  <_  B  /\  B  <_  sup ( ( A O B ) ,  RR* ,  <  ) ) ) )
6726, 7, 66syl2anc 661 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( sup ( ( A O B ) ,  RR* ,  <  )  =  B  <-> 
( sup ( ( A O B ) ,  RR* ,  <  )  <_  B  /\  B  <_  sup ( ( A O B ) ,  RR* ,  <  ) ) ) )
6817, 65, 67mpbir2and 920 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  sup (
( A O B ) ,  RR* ,  <  )  =  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815   {crab 2818    C_ wss 3476   (/)c0 3785   class class class wbr 4447  (class class class)co 6282    |-> cmpt2 6284   supcsup 7896   RR*cxr 9623    < clt 9624    <_ cle 9625   QQcq 11178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-1st 6781  df-2nd 6782  df-recs 7039  df-rdg 7073  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-n0 10792  df-z 10861  df-uz 11079  df-q 11179
This theorem is referenced by:  ioopnfsup  11954  icopnfsup  11955  bndth  21190  ioorf  21714  ioorinv2  21716  ioossioobi  31121
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