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Theorem ixxub 11408
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 11396 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A O B )  <->  ( w  e.  RR*  /\  A R w  /\  w S B ) ) )
323adant3 1008 . . . . . . 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 1002 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  w S B )
64simp1d 1000 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  w  e.  RR* )
7 simp2 989 . . . . . . 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 2881 . . 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 3446 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( A O B )  C_  RR* )
15 supxrleub 11376 . . . 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 11045 . . . . . . . . . . 11  |-  ( w  e.  QQ  ->  w  e.  RR )
2120rexrd 9520 . . . . . . . . . 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 988 . . . . . . . . . . . 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 11364 . . . . . . . . . . . . 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 990 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( A O B )  =/=  (/) )
29 n0 3730 . . . . . . . . . . . . . 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 1001 . . . . . . . . . . . . . . 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 11374 . . . . . . . . . . . . . . 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 11223 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  A  <_  sup ( ( A O B ) , 
RR* ,  <  ) )
4030, 39exlimddv 1693 . . . . . . . . . . . 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 11221 . . . . . . . . . 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 1171 . . . . . . . 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 9529 . . . . . . . 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 2901 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  -.  E. w  e.  QQ  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  <  B ) )
59 qbtwnxr 11257 . . . . . 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 1190 . . . . 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 9529 . . . 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 11216 . . 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 913 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 965    = wceq 1370   E.wex 1587    e. wcel 1757    =/= wne 2641   A.wral 2792   E.wrex 2793   {crab 2796    C_ wss 3412   (/)c0 3721   class class class wbr 4376  (class class class)co 6176    |-> cmpt2 6178   supcsup 7777   RR*cxr 9504    < clt 9505    <_ cle 9506   QQcq 11040
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446  ax-pre-sup 9447
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-1st 6663  df-2nd 6664  df-recs 6918  df-rdg 6952  df-er 7187  df-en 7397  df-dom 7398  df-sdom 7399  df-sup 7778  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-div 10081  df-nn 10410  df-n0 10667  df-z 10734  df-uz 10949  df-q 11041
This theorem is referenced by:  ioopnfsup  11790  icopnfsup  11791  bndth  20632  ioorf  21155  ioorinv2  21157
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