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Theorem ixxub 11656
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 11644 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A O B )  <->  ( w  e.  RR*  /\  A R w  /\  w S B ) ) )
323adant3 1028 . . . . . . 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 487 . . . . . 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 1022 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  w S B )
64simp1d 1020 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  w  e.  RR* )
7 simp2 1009 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  B  e. 
RR* )
87adantr 467 . . . . . 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 667 . . . . 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 2802 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  A. w  e.  ( A O B ) w  <_  B
)
136ex 436 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( w  e.  ( A O B )  ->  w  e.  RR* ) )
1413ssrdv 3438 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( A O B )  C_  RR* )
15 supxrleub 11612 . . . 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 667 . . 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 236 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  sup (
( A O B ) ,  RR* ,  <  )  <_  B )
18 simprl 764 . . . . . 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 732 . . . . . . . 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 11269 . . . . . . . . . . 11  |-  ( w  e.  QQ  ->  w  e.  RR )
2120rexrd 9690 . . . . . . . . . 10  |-  ( w  e.  QQ  ->  w  e.  RR* )
2221ad2antlr 733 . . . . . . . . 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 1008 . . . . . . . . . . . 12  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  A  e. 
RR* )
2423ad2antrr 732 . . . . . . . . . . 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 11600 . . . . . . . . . . . . 13  |-  ( ( A O B ) 
C_  RR*  ->  sup (
( A O B ) ,  RR* ,  <  )  e.  RR* )
2614, 25syl 17 . . . . . . . . . . . 12  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  sup (
( A O B ) ,  RR* ,  <  )  e.  RR* )
2726ad2antrr 732 . . . . . . . . . . 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 1010 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( A O B )  =/=  (/) )
29 n0 3741 . . . . . . . . . . . . . 14  |-  ( ( A O B )  =/=  (/)  <->  E. w  w  e.  ( A O B ) )
3028, 29sylib 200 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  E. w  w  e.  ( A O B ) )
3123adantr 467 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  A  e.  RR* )
3226adantr 467 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  sup ( ( A O B ) ,  RR* ,  <  )  e.  RR* )
334simp2d 1021 . . . . . . . . . . . . . . 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 667 . . . . . . . . . . . . . . 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 11610 . . . . . . . . . . . . . . 15  |-  ( ( ( A O B )  C_  RR*  /\  w  e.  ( A O B ) )  ->  w  <_  sup ( ( A O B ) , 
RR* ,  <  ) )
3814, 37sylan 474 . . . . . . . . . . . . . 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 11459 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  A  <_  sup ( ( A O B ) , 
RR* ,  <  ) )
4030, 39exlimddv 1781 . . . . . . . . . . . 12  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  A  <_  sup ( ( A O B ) ,  RR* ,  <  ) )
4140ad2antrr 732 . . . . . . . . . . 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 11457 . . . . . . . . . 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 667 . . . . . . . . . 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 766 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  < 
B ) )  ->  w  <  B )
477ad2antrr 732 . . . . . . . . . . 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 667 . . . . . . . . . 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 732 . . . . . . . . 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 1191 . . . . . . . 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 667 . . . . . . 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 9699 . . . . . . . 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 667 . . . . . . 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 214 . . . . . 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 580 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  ->  -.  ( sup ( ( A O B ) , 
RR* ,  <  )  < 
w  /\  w  <  B ) )
5857nrexdv 2843 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  -.  E. w  e.  QQ  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  <  B ) )
59 qbtwnxr 11493 . . . . . 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 1210 . . . . 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 667 . . . 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 181 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  -.  sup ( ( A O B ) ,  RR* ,  <  )  <  B
)
63 xrlenlt 9699 . . . 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 667 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( B  <_  sup ( ( A O B ) , 
RR* ,  <  )  <->  -.  sup (
( A O B ) ,  RR* ,  <  )  <  B ) )
6562, 64mpbird 236 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  B  <_  sup ( ( A O B ) ,  RR* ,  <  ) )
66 xrletri3 11451 . . 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 667 . 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 933 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 188    /\ wa 371    /\ w3a 985    = wceq 1444   E.wex 1663    e. wcel 1887    =/= wne 2622   A.wral 2737   E.wrex 2738   {crab 2741    C_ wss 3404   (/)c0 3731   class class class wbr 4402  (class class class)co 6290    |-> cmpt2 6292   supcsup 7954   RR*cxr 9674    < clt 9675    <_ cle 9676   QQcq 11264
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616  ax-pre-sup 9617
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-sup 7956  df-inf 7957  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-q 11265
This theorem is referenced by:  ioopnfsup  12091  icopnfsup  12092  bndth  21986  ioorf  22525  ioorinv2  22527  ioorfOLD  22530  ioorinv2OLD  22532  ioossioobi  37618
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