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Theorem ixxlb 11472
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 755 . . . . . 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 11459 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A O B )  <->  ( w  e.  RR*  /\  A R w  /\  w S B ) ) )
433adant3 1014 . . . . . . . . . . . . 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 482 . . . . . . . . . . . 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 1006 . . . . . . . . . . 11  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  w  e.  RR* )
76ex 432 . . . . . . . . . 10  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( w  e.  ( A O B )  ->  w  e.  RR* ) )
87ssrdv 3423 . . . . . . . . 9  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( A O B )  C_  RR* )
98ad2antrr 723 . . . . . . . 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 11106 . . . . . . . . . . 11  |-  ( w  e.  QQ  ->  w  e.  RR )
1110rexrd 9554 . . . . . . . . . 10  |-  ( w  e.  QQ  ->  w  e.  RR* )
1211ad2antlr 724 . . . . . . . . 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 754 . . . . . . . . . 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 994 . . . . . . . . . . . 12  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  A  e. 
RR* )
1514ad2antrr 723 . . . . . . . . . . 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 659 . . . . . . . . . 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 11429 . . . . . . . . . . . . 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 723 . . . . . . . . . . 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 1034 . . . . . . . . . . 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 996 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( A O B )  =/=  (/) )
24 n0 3721 . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  sup ( ( A O B ) ,  RR* ,  `'  <  )  e.  RR* )
27 simpl2 998 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  B  e.  RR* )
28 infmxrlb 11446 . . . . . . . . . . . . . . 15  |-  ( ( ( A O B )  C_  RR*  /\  w  e.  ( A O B ) )  ->  sup ( ( A O B ) ,  RR* ,  `'  <  )  <_  w
)
298, 28sylan 469 . . . . . . . . . . . . . 14  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  sup ( ( A O B ) ,  RR* ,  `'  <  )  <_  w
)
305simp3d 1008 . . . . . . . . . . . . . . 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 659 . . . . . . . . . . . . . . 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 11286 . . . . . . . . . . . . 13  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  sup ( ( A O B ) ,  RR* ,  `'  <  )  <_  B
)
3525, 34exlimddv 1734 . . . . . . . . . . . 12  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  sup (
( A O B ) ,  RR* ,  `'  <  )  <_  B )
3635ad2antrr 723 . . . . . . . . . . 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 11285 . . . . . . . . . 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 659 . . . . . . . . . 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 723 . . . . . . . . 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 1177 . . . . . . . 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 659 . . . . . . 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 9563 . . . . . . . 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 659 . . . . . . 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 574 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  ->  -.  ( A  <  w  /\  w  <  sup ( ( A O B ) , 
RR* ,  `'  <  ) ) )
4847nrexdv 2838 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  -.  E. w  e.  QQ  ( A  <  w  /\  w  <  sup ( ( A O B ) , 
RR* ,  `'  <  ) ) )
49 qbtwnxr 11320 . . . . . 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 1196 . . . . 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 659 . . . 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 9563 . . . 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 659 . . 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 1007 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  A R w )
5714adantr 463 . . . . . 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 659 . . . . 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 2796 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  A. w  e.  ( A O B ) A  <_  w
)
62 infmxrgelb 11447 . . . 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 659 . . 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 11279 . . 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 659 . 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 920 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 367    /\ w3a 971    = wceq 1399   E.wex 1620    e. wcel 1826    =/= wne 2577   A.wral 2732   E.wrex 2733   {crab 2736    C_ wss 3389   (/)c0 3711   class class class wbr 4367   `'ccnv 4912  (class class class)co 6196    |-> cmpt2 6198   supcsup 7815   RR*cxr 9538    < clt 9539    <_ cle 9540   QQcq 11101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601  ax-un 6491  ax-cnex 9459  ax-resscn 9460  ax-1cn 9461  ax-icn 9462  ax-addcl 9463  ax-addrcl 9464  ax-mulcl 9465  ax-mulrcl 9466  ax-mulcom 9467  ax-addass 9468  ax-mulass 9469  ax-distr 9470  ax-i2m1 9471  ax-1ne0 9472  ax-1rid 9473  ax-rnegex 9474  ax-rrecex 9475  ax-cnre 9476  ax-pre-lttri 9477  ax-pre-lttrn 9478  ax-pre-ltadd 9479  ax-pre-mulgt0 9480  ax-pre-sup 9481
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-nel 2580  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-csb 3349  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-pss 3405  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-tp 3949  df-op 3951  df-uni 4164  df-iun 4245  df-br 4368  df-opab 4426  df-mpt 4427  df-tr 4461  df-eprel 4705  df-id 4709  df-po 4714  df-so 4715  df-fr 4752  df-we 4754  df-ord 4795  df-on 4796  df-lim 4797  df-suc 4798  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-rn 4924  df-res 4925  df-ima 4926  df-iota 5460  df-fun 5498  df-fn 5499  df-f 5500  df-f1 5501  df-fo 5502  df-f1o 5503  df-fv 5504  df-riota 6158  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-om 6600  df-1st 6699  df-2nd 6700  df-recs 6960  df-rdg 6994  df-er 7229  df-en 7436  df-dom 7437  df-sdom 7438  df-sup 7816  df-pnf 9541  df-mnf 9542  df-xr 9543  df-ltxr 9544  df-le 9545  df-sub 9720  df-neg 9721  df-div 10124  df-nn 10453  df-n0 10713  df-z 10782  df-uz 11002  df-q 11102
This theorem is referenced by:  ioorf  22067  ioorinv2  22069  ioossioobi  31723
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