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Theorem ixxlb 11542
Description: Extract the lower 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
ixxlb  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  sup (
( A O B ) ,  RR* ,  `'  <  )  =  A )
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 ixxlb
StepHypRef Expression
1 simprr 756 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  ->  w  <  sup ( ( A O B ) , 
RR* ,  `'  <  ) )
2 ixx.1 . . . . . . . . . . . . . . 15  |-  O  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x R z  /\  z S y ) } )
32elixx1 11529 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A O B )  <->  ( w  e.  RR*  /\  A R w  /\  w S B ) ) )
433adant3 1011 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( w  e.  ( A O B )  <->  ( w  e.  RR*  /\  A R w  /\  w S B ) ) )
54biimpa 484 . . . . . . . . . . . 12  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  (
w  e.  RR*  /\  A R w  /\  w S B ) )
65simp1d 1003 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  w  e.  RR* )
76ex 434 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( w  e.  ( A O B )  ->  w  e.  RR* ) )
87ssrdv 3505 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( A O B )  C_  RR* )
98ad2antrr 725 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  -> 
( A O B )  C_  RR* )
10 qre 11178 . . . . . . . . . . 11  |-  ( w  e.  QQ  ->  w  e.  RR )
1110rexrd 9634 . . . . . . . . . 10  |-  ( w  e.  QQ  ->  w  e.  RR* )
1211ad2antlr 726 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  ->  w  e.  RR* )
13 simprl 755 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  ->  A  <  w )
14 simp1 991 . . . . . . . . . . . 12  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  A  e. 
RR* )
1514ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  ->  A  e.  RR* )
16 ixxub.4 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  ->  A R w ) )
1715, 12, 16syl2anc 661 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  -> 
( A  <  w  ->  A R w ) )
1813, 17mpd 15 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  ->  A R w )
19 infmxrcl 11499 . . . . . . . . . . . . 13  |-  ( ( A O B ) 
C_  RR*  ->  sup (
( A O B ) ,  RR* ,  `'  <  )  e.  RR* )
208, 19syl 16 . . . . . . . . . . . 12  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  sup (
( A O B ) ,  RR* ,  `'  <  )  e.  RR* )
2120ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  ->  sup ( ( A O B ) ,  RR* ,  `'  <  )  e.  RR* )
22 simpll2 1031 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  ->  B  e.  RR* )
23 simp3 993 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( A O B )  =/=  (/) )
24 n0 3789 . . . . . . . . . . . . . 14  |-  ( ( A O B )  =/=  (/)  <->  E. w  w  e.  ( A O B ) )
2523, 24sylib 196 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  E. w  w  e.  ( A O B ) )
2620adantr 465 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  sup ( ( A O B ) ,  RR* ,  `'  <  )  e.  RR* )
27 simpl2 995 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  B  e.  RR* )
28 infmxrlb 11516 . . . . . . . . . . . . . . 15  |-  ( ( ( A O B )  C_  RR*  /\  w  e.  ( A O B ) )  ->  sup ( ( A O B ) ,  RR* ,  `'  <  )  <_  w
)
298, 28sylan 471 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  sup ( ( A O B ) ,  RR* ,  `'  <  )  <_  w
)
305simp3d 1005 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  w S B )
31 ixxub.3 . . . . . . . . . . . . . . . 16  |-  ( ( w  e.  RR*  /\  B  e.  RR* )  ->  (
w S B  ->  w  <_  B ) )
326, 27, 31syl2anc 661 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  (
w S B  ->  w  <_  B ) )
3330, 32mpd 15 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  w  <_  B )
3426, 6, 27, 29, 33xrletrd 11356 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  sup ( ( A O B ) ,  RR* ,  `'  <  )  <_  B
)
3525, 34exlimddv 1697 . . . . . . . . . . . 12  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  sup (
( A O B ) ,  RR* ,  `'  <  )  <_  B )
3635ad2antrr 725 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  ->  sup ( ( A O B ) ,  RR* ,  `'  <  )  <_  B
)
3712, 21, 22, 1, 36xrltletrd 11355 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  ->  w  <  B )
38 ixxub.2 . . . . . . . . . . 11  |-  ( ( w  e.  RR*  /\  B  e.  RR* )  ->  (
w  <  B  ->  w S B ) )
3912, 22, 38syl2anc 661 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  -> 
( w  <  B  ->  w S B ) )
4037, 39mpd 15 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  ->  w S B )
414ad2antrr 725 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  -> 
( w  e.  ( A O B )  <-> 
( w  e.  RR*  /\  A R w  /\  w S B ) ) )
4212, 18, 40, 41mpbir3and 1174 . . . . . . . 8  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  ->  w  e.  ( A O B ) )
439, 42, 28syl2anc 661 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  ->  sup ( ( A O B ) ,  RR* ,  `'  <  )  <_  w
)
44 xrlenlt 9643 . . . . . . . 8  |-  ( ( sup ( ( A O B ) , 
RR* ,  `'  <  )  e.  RR*  /\  w  e.  RR* )  ->  ( sup ( ( A O B ) ,  RR* ,  `'  <  )  <_  w  <->  -.  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )
4521, 12, 44syl2anc 661 . . . . . . 7  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  -> 
( sup ( ( A O B ) ,  RR* ,  `'  <  )  <_  w  <->  -.  w  <  sup ( ( A O B ) , 
RR* ,  `'  <  ) ) )
4643, 45mpbid 210 . . . . . 6  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( A  <  w  /\  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )  ->  -.  w  <  sup (
( A O B ) ,  RR* ,  `'  <  ) )
471, 46pm2.65da 576 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  ->  -.  ( A  <  w  /\  w  <  sup ( ( A O B ) , 
RR* ,  `'  <  ) ) )
4847nrexdv 2915 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  -.  E. w  e.  QQ  ( A  <  w  /\  w  <  sup ( ( A O B ) , 
RR* ,  `'  <  ) ) )
49 qbtwnxr 11390 . . . . . 6  |-  ( ( A  e.  RR*  /\  sup ( ( A O B ) ,  RR* ,  `'  <  )  e.  RR*  /\  A  <  sup (
( A O B ) ,  RR* ,  `'  <  ) )  ->  E. w  e.  QQ  ( A  < 
w  /\  w  <  sup ( ( A O B ) ,  RR* ,  `'  <  ) ) )
50493expia 1193 . . . . 5  |-  ( ( A  e.  RR*  /\  sup ( ( A O B ) ,  RR* ,  `'  <  )  e.  RR* )  ->  ( A  <  sup ( ( A O B ) ,  RR* ,  `'  <  )  ->  E. w  e.  QQ  ( A  < 
w  /\  w  <  sup ( ( A O B ) ,  RR* ,  `'  <  ) ) ) )
5114, 20, 50syl2anc 661 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( A  <  sup ( ( A O B ) , 
RR* ,  `'  <  )  ->  E. w  e.  QQ  ( A  <  w  /\  w  <  sup ( ( A O B ) , 
RR* ,  `'  <  ) ) ) )
5248, 51mtod 177 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  -.  A  <  sup ( ( A O B ) , 
RR* ,  `'  <  ) )
53 xrlenlt 9643 . . . 4  |-  ( ( sup ( ( A O B ) , 
RR* ,  `'  <  )  e.  RR*  /\  A  e. 
RR* )  ->  ( sup ( ( A O B ) ,  RR* ,  `'  <  )  <_  A  <->  -.  A  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )
5420, 14, 53syl2anc 661 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( sup ( ( A O B ) ,  RR* ,  `'  <  )  <_  A  <->  -.  A  <  sup (
( A O B ) ,  RR* ,  `'  <  ) ) )
5552, 54mpbird 232 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  sup (
( A O B ) ,  RR* ,  `'  <  )  <_  A )
565simp2d 1004 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  A R w )
5714adantr 465 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  A  e.  RR* )
58 ixxub.5 . . . . . 6  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A R w  ->  A  <_  w ) )
5957, 6, 58syl2anc 661 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  ( A R w  ->  A  <_  w ) )
6056, 59mpd 15 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  A  <_  w )
6160ralrimiva 2873 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  A. w  e.  ( A O B ) A  <_  w
)
62 infmxrgelb 11517 . . . 4  |-  ( ( ( A O B )  C_  RR*  /\  A  e.  RR* )  ->  ( A  <_  sup ( ( A O B ) , 
RR* ,  `'  <  )  <->  A. w  e.  ( A O B ) A  <_  w ) )
638, 14, 62syl2anc 661 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( A  <_  sup ( ( A O B ) , 
RR* ,  `'  <  )  <->  A. w  e.  ( A O B ) A  <_  w ) )
6461, 63mpbird 232 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  A  <_  sup ( ( A O B ) ,  RR* ,  `'  <  ) )
65 xrletri3 11349 . . 3  |-  ( ( sup ( ( A O B ) , 
RR* ,  `'  <  )  e.  RR*  /\  A  e. 
RR* )  ->  ( sup ( ( A O B ) ,  RR* ,  `'  <  )  =  A  <-> 
( sup ( ( A O B ) ,  RR* ,  `'  <  )  <_  A  /\  A  <_  sup ( ( A O B ) , 
RR* ,  `'  <  ) ) ) )
6620, 14, 65syl2anc 661 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( sup ( ( A O B ) ,  RR* ,  `'  <  )  =  A  <-> 
( sup ( ( A O B ) ,  RR* ,  `'  <  )  <_  A  /\  A  <_  sup ( ( A O B ) , 
RR* ,  `'  <  ) ) ) )
6755, 64, 66mpbir2and 915 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  sup (
( A O B ) ,  RR* ,  `'  <  )  =  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 968    = wceq 1374   E.wex 1591    e. wcel 1762    =/= wne 2657   A.wral 2809   E.wrex 2810   {crab 2813    C_ wss 3471   (/)c0 3780   class class class wbr 4442   `'ccnv 4993  (class class class)co 6277    |-> cmpt2 6279   supcsup 7891   RR*cxr 9618    < clt 9619    <_ cle 9620   QQcq 11173
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pow 4620  ax-pr 4681  ax-un 6569  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-nel 2660  df-ral 2814  df-rex 2815  df-reu 2816  df-rmo 2817  df-rab 2818  df-v 3110  df-sbc 3327  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3781  df-if 3935  df-pw 4007  df-sn 4023  df-pr 4025  df-tp 4027  df-op 4029  df-uni 4241  df-iun 4322  df-br 4443  df-opab 4501  df-mpt 4502  df-tr 4536  df-eprel 4786  df-id 4790  df-po 4795  df-so 4796  df-fr 4833  df-we 4835  df-ord 4876  df-on 4877  df-lim 4878  df-suc 4879  df-xp 5000  df-rel 5001  df-cnv 5002  df-co 5003  df-dm 5004  df-rn 5005  df-res 5006  df-ima 5007  df-iota 5544  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6674  df-1st 6776  df-2nd 6777  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7509  df-dom 7510  df-sdom 7511  df-sup 7892  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9798  df-neg 9799  df-div 10198  df-nn 10528  df-n0 10787  df-z 10856  df-uz 11074  df-q 11174
This theorem is referenced by:  ioorf  21712  ioorinv2  21714  ioossioobi  31078
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