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Theorem ixxub 10677
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 10665 . . . . . . . 8  |-  ( ( A  e.  RR*  /\  B  e.  RR* )  ->  (
w  e.  ( A O B )  <->  ( w  e.  RR*  /\  A R w  /\  w S B ) ) )
323adant3 975 . . . . . . 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 470 . . . . . 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 969 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  w S B )
64simp1d 967 . . . . . 6  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  w  e.  RR* )
7 simp2 956 . . . . . . 7  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  B  e. 
RR* )
87adantr 451 . . . . . 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 642 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  (
w S B  ->  w  <_  B ) )
115, 10mpd 14 . . . 4  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  w  <_  B )
1211ralrimiva 2626 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  A. w  e.  ( A O B ) w  <_  B
)
136ex 423 . . . . 5  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( w  e.  ( A O B )  ->  w  e.  RR* ) )
1413ssrdv 3185 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( A O B )  C_  RR* )
15 supxrleub 10645 . . . 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 642 . . 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 223 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  sup (
( A O B ) ,  RR* ,  <  )  <_  B )
18 simprl 732 . . . . . 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 706 . . . . . . . 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 10321 . . . . . . . . . . 11  |-  ( w  e.  QQ  ->  w  e.  RR )
2120rexrd 8881 . . . . . . . . . 10  |-  ( w  e.  QQ  ->  w  e.  RR* )
2221ad2antlr 707 . . . . . . . . 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 955 . . . . . . . . . . . 12  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  A  e. 
RR* )
2423ad2antrr 706 . . . . . . . . . . 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 10633 . . . . . . . . . . . . 13  |-  ( ( A O B ) 
C_  RR*  ->  sup (
( A O B ) ,  RR* ,  <  )  e.  RR* )
2614, 25syl 15 . . . . . . . . . . . 12  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  sup (
( A O B ) ,  RR* ,  <  )  e.  RR* )
2726ad2antrr 706 . . . . . . . . . . 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 957 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( A O B )  =/=  (/) )
29 n0 3464 . . . . . . . . . . . . . 14  |-  ( ( A O B )  =/=  (/)  <->  E. w  w  e.  ( A O B ) )
3028, 29sylib 188 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  E. w  w  e.  ( A O B ) )
3123adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  A  e.  RR* )
3226adantr 451 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  sup ( ( A O B ) ,  RR* ,  <  )  e.  RR* )
334simp2d 968 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  A R w )
34 ixxub.5 . . . . . . . . . . . . . . . . . 18  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A R w  ->  A  <_  w ) )
3531, 6, 34syl2anc 642 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  ( A R w  ->  A  <_  w ) )
3633, 35mpd 14 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  A  <_  w )
37 supxrub 10643 . . . . . . . . . . . . . . . . 17  |-  ( ( ( A O B )  C_  RR*  /\  w  e.  ( A O B ) )  ->  w  <_  sup ( ( A O B ) , 
RR* ,  <  ) )
3814, 37sylan 457 . . . . . . . . . . . . . . . 16  |-  ( ( ( 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 10493 . . . . . . . . . . . . . . 15  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  ( A O B ) )  ->  A  <_  sup ( ( A O B ) , 
RR* ,  <  ) )
4039ex 423 . . . . . . . . . . . . . 14  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( w  e.  ( A O B )  ->  A  <_  sup ( ( A O B ) , 
RR* ,  <  ) ) )
4140exlimdv 1664 . . . . . . . . . . . . 13  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( E. w  w  e.  ( A O B )  ->  A  <_  sup ( ( A O B ) ,  RR* ,  <  ) ) )
4230, 41mpd 14 . . . . . . . . . . . 12  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  A  <_  sup ( ( A O B ) ,  RR* ,  <  ) )
4342ad2antrr 706 . . . . . . . . . . 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* ,  <  ) )
4424, 27, 22, 43, 18xrlelttrd 10491 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  < 
B ) )  ->  A  <  w )
45 ixxub.4 . . . . . . . . . . 11  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  ->  A R w ) )
4624, 22, 45syl2anc 642 . . . . . . . . . 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 ) )
4744, 46mpd 14 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  < 
B ) )  ->  A R w )
48 simprr 733 . . . . . . . . . 10  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  < 
B ) )  ->  w  <  B )
497ad2antrr 706 . . . . . . . . . . 11  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  < 
B ) )  ->  B  e.  RR* )
50 ixxub.2 . . . . . . . . . . 11  |-  ( ( w  e.  RR*  /\  B  e.  RR* )  ->  (
w  <  B  ->  w S B ) )
5122, 49, 50syl2anc 642 . . . . . . . . . 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 ) )
5248, 51mpd 14 . . . . . . . . 9  |-  ( ( ( ( A  e. 
RR*  /\  B  e.  RR* 
/\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  /\  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  < 
B ) )  ->  w S B )
533ad2antrr 706 . . . . . . . . 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 ) ) )
5422, 47, 52, 53mpbir3and 1135 . . . . . . . 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 ) )
5519, 54, 37syl2anc 642 . . . . . . 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* ,  <  ) )
56 xrlenlt 8890 . . . . . . . 8  |-  ( ( w  e.  RR*  /\  sup ( ( A O B ) ,  RR* ,  <  )  e.  RR* )  ->  ( w  <_  sup ( ( A O B ) ,  RR* ,  <  )  <->  -.  sup (
( A O B ) ,  RR* ,  <  )  <  w ) )
5722, 27, 56syl2anc 642 . . . . . . 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 ) )
5855, 57mpbid 201 . . . . . 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 )
5918, 58pm2.65da 559 . . . . 5  |-  ( ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  /\  w  e.  QQ )  ->  -.  ( sup ( ( A O B ) , 
RR* ,  <  )  < 
w  /\  w  <  B ) )
6059nrexdv 2646 . . . 4  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  -.  E. w  e.  QQ  ( sup ( ( A O B ) ,  RR* ,  <  )  <  w  /\  w  <  B ) )
61 qbtwnxr 10527 . . . . . 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 ) )
62613expia 1153 . . . . 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 ) ) )
6326, 7, 62syl2anc 642 . . . 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 ) ) )
6460, 63mtod 168 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  -.  sup ( ( A O B ) ,  RR* ,  <  )  <  B
)
65 xrlenlt 8890 . . . 4  |-  ( ( B  e.  RR*  /\  sup ( ( A O B ) ,  RR* ,  <  )  e.  RR* )  ->  ( B  <_  sup ( ( A O B ) ,  RR* ,  <  )  <->  -.  sup (
( A O B ) ,  RR* ,  <  )  <  B ) )
667, 26, 65syl2anc 642 . . 3  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  ( B  <_  sup ( ( A O B ) , 
RR* ,  <  )  <->  -.  sup (
( A O B ) ,  RR* ,  <  )  <  B ) )
6764, 66mpbird 223 . 2  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  B  <_  sup ( ( A O B ) ,  RR* ,  <  ) )
68 xrletri3 10486 . . 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* ,  <  ) ) ) )
6926, 7, 68syl2anc 642 . 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* ,  <  ) ) ) )
7017, 67, 69mpbir2and 888 1  |-  ( ( A  e.  RR*  /\  B  e.  RR*  /\  ( A O B )  =/=  (/) )  ->  sup (
( A O B ) ,  RR* ,  <  )  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544   {crab 2547    C_ wss 3152   (/)c0 3455   class class class wbr 4023  (class class class)co 5858    e. cmpt2 5860   supcsup 7193   RR*cxr 8866    < clt 8867    <_ cle 8868   QQcq 10316
This theorem is referenced by:  ioopnfsup  10968  icopnfsup  10969  bndth  18456  ioorf  18928  ioorinv2  18930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-q 10317
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