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Theorem supxrbnd 10832
Description: The supremum of a bounded-above nonempty set of reals is real. (Contributed by NM, 19-Jan-2006.)
Assertion
Ref Expression
supxrbnd  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  sup ( A ,  RR* ,  <  )  <  +oo )  ->  sup ( A ,  RR* ,  <  )  e.  RR )

Proof of Theorem supxrbnd
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ressxr 9055 . . . . 5  |-  RR  C_  RR*
2 sstr 3292 . . . . 5  |-  ( ( A  C_  RR  /\  RR  C_ 
RR* )  ->  A  C_ 
RR* )
31, 2mpan2 653 . . . 4  |-  ( A 
C_  RR  ->  A  C_  RR* )
4 supxrcl 10818 . . . . . . 7  |-  ( A 
C_  RR*  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
5 pnfxr 10638 . . . . . . . . . 10  |-  +oo  e.  RR*
6 xrltne 10678 . . . . . . . . . 10  |-  ( ( sup ( A ,  RR* ,  <  )  e. 
RR*  /\  +oo  e.  RR*  /\ 
sup ( A ,  RR* ,  <  )  <  +oo )  ->  +oo  =/=  sup ( A ,  RR* ,  <  ) )
75, 6mp3an2 1267 . . . . . . . . 9  |-  ( ( sup ( A ,  RR* ,  <  )  e. 
RR*  /\  sup ( A ,  RR* ,  <  )  <  +oo )  ->  +oo  =/=  sup ( A ,  RR* ,  <  ) )
87necomd 2626 . . . . . . . 8  |-  ( ( sup ( A ,  RR* ,  <  )  e. 
RR*  /\  sup ( A ,  RR* ,  <  )  <  +oo )  ->  sup ( A ,  RR* ,  <  )  =/=  +oo )
98ex 424 . . . . . . 7  |-  ( sup ( A ,  RR* ,  <  )  e.  RR*  ->  ( sup ( A ,  RR* ,  <  )  <  +oo  ->  sup ( A ,  RR* ,  <  )  =/=  +oo ) )
104, 9syl 16 . . . . . 6  |-  ( A 
C_  RR*  ->  ( sup ( A ,  RR* ,  <  )  <  +oo  ->  sup ( A ,  RR* ,  <  )  =/=  +oo ) )
11 supxrunb2 10824 . . . . . . . . 9  |-  ( A 
C_  RR*  ->  ( A. x  e.  RR  E. y  e.  A  x  <  y  <->  sup ( A ,  RR* ,  <  )  =  +oo ) )
12 ssel2 3279 . . . . . . . . . . . . . 14  |-  ( ( A  C_  RR*  /\  y  e.  A )  ->  y  e.  RR* )
1312adantlr 696 . . . . . . . . . . . . 13  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  y  e.  A )  ->  y  e.  RR* )
14 rexr 9056 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  x  e.  RR* )
1514ad2antlr 708 . . . . . . . . . . . . 13  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  y  e.  A )  ->  x  e.  RR* )
16 xrlenlt 9069 . . . . . . . . . . . . . 14  |-  ( ( y  e.  RR*  /\  x  e.  RR* )  ->  (
y  <_  x  <->  -.  x  <  y ) )
1716con2bid 320 . . . . . . . . . . . . 13  |-  ( ( y  e.  RR*  /\  x  e.  RR* )  ->  (
x  <  y  <->  -.  y  <_  x ) )
1813, 15, 17syl2anc 643 . . . . . . . . . . . 12  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  y  e.  A )  ->  (
x  <  y  <->  -.  y  <_  x ) )
1918rexbidva 2659 . . . . . . . . . . 11  |-  ( ( A  C_  RR*  /\  x  e.  RR )  ->  ( E. y  e.  A  x  <  y  <->  E. y  e.  A  -.  y  <_  x ) )
20 rexnal 2653 . . . . . . . . . . 11  |-  ( E. y  e.  A  -.  y  <_  x  <->  -.  A. y  e.  A  y  <_  x )
2119, 20syl6bb 253 . . . . . . . . . 10  |-  ( ( A  C_  RR*  /\  x  e.  RR )  ->  ( E. y  e.  A  x  <  y  <->  -.  A. y  e.  A  y  <_  x ) )
2221ralbidva 2658 . . . . . . . . 9  |-  ( A 
C_  RR*  ->  ( A. x  e.  RR  E. y  e.  A  x  <  y  <->  A. x  e.  RR  -.  A. y  e.  A  y  <_  x ) )
2311, 22bitr3d 247 . . . . . . . 8  |-  ( A 
C_  RR*  ->  ( sup ( A ,  RR* ,  <  )  =  +oo  <->  A. x  e.  RR  -.  A. y  e.  A  y  <_  x ) )
24 ralnex 2652 . . . . . . . 8  |-  ( A. x  e.  RR  -.  A. y  e.  A  y  <_  x  <->  -.  E. x  e.  RR  A. y  e.  A  y  <_  x
)
2523, 24syl6bb 253 . . . . . . 7  |-  ( A 
C_  RR*  ->  ( sup ( A ,  RR* ,  <  )  =  +oo  <->  -.  E. x  e.  RR  A. y  e.  A  y  <_  x
) )
2625necon2abid 2600 . . . . . 6  |-  ( A 
C_  RR*  ->  ( E. x  e.  RR  A. y  e.  A  y  <_  x  <->  sup ( A ,  RR* ,  <  )  =/=  +oo ) )
2710, 26sylibrd 226 . . . . 5  |-  ( A 
C_  RR*  ->  ( sup ( A ,  RR* ,  <  )  <  +oo  ->  E. x  e.  RR  A. y  e.  A  y  <_  x
) )
2827imp 419 . . . 4  |-  ( ( A  C_  RR*  /\  sup ( A ,  RR* ,  <  )  <  +oo )  ->  E. x  e.  RR  A. y  e.  A  y  <_  x
)
293, 28sylan 458 . . 3  |-  ( ( A  C_  RR  /\  sup ( A ,  RR* ,  <  )  <  +oo )  ->  E. x  e.  RR  A. y  e.  A  y  <_  x
)
30293adant2 976 . 2  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  sup ( A ,  RR* ,  <  )  <  +oo )  ->  E. x  e.  RR  A. y  e.  A  y  <_  x
)
31 supxrre 10831 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR* ,  <  )  =  sup ( A ,  RR ,  <  ) )
32 suprcl 9893 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR ,  <  )  e.  RR )
3331, 32eqeltrd 2454 . 2  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR* ,  <  )  e.  RR )
3430, 33syld3an3 1229 1  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  sup ( A ,  RR* ,  <  )  <  +oo )  ->  sup ( A ,  RR* ,  <  )  e.  RR )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717    =/= wne 2543   A.wral 2642   E.wrex 2643    C_ wss 3256   (/)c0 3564   class class class wbr 4146   supcsup 7373   RRcr 8915    +oocpnf 9043   RR*cxr 9045    < clt 9046    <_ cle 9047
This theorem is referenced by:  supxrgtmnf  10833  ovolunlem1  19253  uniioombllem1  19333
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634  ax-cnex 8972  ax-resscn 8973  ax-1cn 8974  ax-icn 8975  ax-addcl 8976  ax-addrcl 8977  ax-mulcl 8978  ax-mulrcl 8979  ax-mulcom 8980  ax-addass 8981  ax-mulass 8982  ax-distr 8983  ax-i2m1 8984  ax-1ne0 8985  ax-1rid 8986  ax-rnegex 8987  ax-rrecex 8988  ax-cnre 8989  ax-pre-lttri 8990  ax-pre-lttrn 8991  ax-pre-ltadd 8992  ax-pre-mulgt0 8993  ax-pre-sup 8994
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rmo 2650  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-po 4437  df-so 4438  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-riota 6478  df-er 6834  df-en 7039  df-dom 7040  df-sdom 7041  df-sup 7374  df-pnf 9048  df-mnf 9049  df-xr 9050  df-ltxr 9051  df-le 9052  df-sub 9218  df-neg 9219
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