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Theorem supxrbnd 10899
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 9121 . . . . 5  |-  RR  C_  RR*
2 sstr 3348 . . . . 5  |-  ( ( A  C_  RR  /\  RR  C_ 
RR* )  ->  A  C_ 
RR* )
31, 2mpan2 653 . . . 4  |-  ( A 
C_  RR  ->  A  C_  RR* )
4 supxrcl 10885 . . . . . . 7  |-  ( A 
C_  RR*  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
5 pnfxr 10705 . . . . . . . . . 10  |-  +oo  e.  RR*
6 xrltne 10745 . . . . . . . . . 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 2681 . . . . . . . 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 10891 . . . . . . . . 9  |-  ( A 
C_  RR*  ->  ( A. x  e.  RR  E. y  e.  A  x  <  y  <->  sup ( A ,  RR* ,  <  )  =  +oo ) )
12 ssel2 3335 . . . . . . . . . . . . . 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 9122 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  x  e.  RR* )
1514ad2antlr 708 . . . . . . . . . . . . 13  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  y  e.  A )  ->  x  e.  RR* )
16 xrlenlt 9135 . . . . . . . . . . . . . 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 2714 . . . . . . . . . . 11  |-  ( ( A  C_  RR*  /\  x  e.  RR )  ->  ( E. y  e.  A  x  <  y  <->  E. y  e.  A  -.  y  <_  x ) )
20 rexnal 2708 . . . . . . . . . . 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 2713 . . . . . . . . 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 2707 . . . . . . . 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 2655 . . . . . 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 10898 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR* ,  <  )  =  sup ( A ,  RR ,  <  ) )
32 suprcl 9960 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR ,  <  )  e.  RR )
3331, 32eqeltrd 2509 . 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 1652    e. wcel 1725    =/= wne 2598   A.wral 2697   E.wrex 2698    C_ wss 3312   (/)c0 3620   class class class wbr 4204   supcsup 7437   RRcr 8981    +oocpnf 9109   RR*cxr 9111    < clt 9112    <_ cle 9113
This theorem is referenced by:  supxrgtmnf  10900  ovolunlem1  19385  uniioombllem1  19465
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-po 4495  df-so 4496  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-riota 6541  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-sup 7438  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286
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