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Theorem supxrbnd 11531
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 9640 . . . . 5  |-  RR  C_  RR*
2 sstr 3497 . . . . 5  |-  ( ( A  C_  RR  /\  RR  C_ 
RR* )  ->  A  C_ 
RR* )
31, 2mpan2 671 . . . 4  |-  ( A 
C_  RR  ->  A  C_  RR* )
4 supxrcl 11517 . . . . . . 7  |-  ( A 
C_  RR*  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
5 pnfxr 11332 . . . . . . . . . 10  |- +oo  e.  RR*
6 xrltne 11377 . . . . . . . . . 10  |-  ( ( sup ( A ,  RR* ,  <  )  e. 
RR*  /\ +oo  e.  RR*  /\ 
sup ( A ,  RR* ,  <  )  < +oo )  -> +oo  =/=  sup ( A ,  RR* ,  <  ) )
75, 6mp3an2 1313 . . . . . . . . 9  |-  ( ( sup ( A ,  RR* ,  <  )  e. 
RR*  /\  sup ( A ,  RR* ,  <  )  < +oo )  -> +oo  =/=  sup ( A ,  RR* ,  <  ) )
87necomd 2714 . . . . . . . 8  |-  ( ( sup ( A ,  RR* ,  <  )  e. 
RR*  /\  sup ( A ,  RR* ,  <  )  < +oo )  ->  sup ( A ,  RR* ,  <  )  =/= +oo )
98ex 434 . . . . . . 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 11523 . . . . . . . . 9  |-  ( A 
C_  RR*  ->  ( A. x  e.  RR  E. y  e.  A  x  <  y  <->  sup ( A ,  RR* ,  <  )  = +oo ) )
12 ssel2 3484 . . . . . . . . . . . . . 14  |-  ( ( A  C_  RR*  /\  y  e.  A )  ->  y  e.  RR* )
1312adantlr 714 . . . . . . . . . . . . 13  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  y  e.  A )  ->  y  e.  RR* )
14 rexr 9642 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  x  e.  RR* )
1514ad2antlr 726 . . . . . . . . . . . . 13  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  y  e.  A )  ->  x  e.  RR* )
16 xrlenlt 9655 . . . . . . . . . . . . . 14  |-  ( ( y  e.  RR*  /\  x  e.  RR* )  ->  (
y  <_  x  <->  -.  x  <  y ) )
1716con2bid 329 . . . . . . . . . . . . 13  |-  ( ( y  e.  RR*  /\  x  e.  RR* )  ->  (
x  <  y  <->  -.  y  <_  x ) )
1813, 15, 17syl2anc 661 . . . . . . . . . . . 12  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  y  e.  A )  ->  (
x  <  y  <->  -.  y  <_  x ) )
1918rexbidva 2951 . . . . . . . . . . 11  |-  ( ( A  C_  RR*  /\  x  e.  RR )  ->  ( E. y  e.  A  x  <  y  <->  E. y  e.  A  -.  y  <_  x ) )
20 rexnal 2891 . . . . . . . . . . 11  |-  ( E. y  e.  A  -.  y  <_  x  <->  -.  A. y  e.  A  y  <_  x )
2119, 20syl6bb 261 . . . . . . . . . 10  |-  ( ( A  C_  RR*  /\  x  e.  RR )  ->  ( E. y  e.  A  x  <  y  <->  -.  A. y  e.  A  y  <_  x ) )
2221ralbidva 2879 . . . . . . . . 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 255 . . . . . . . 8  |-  ( A 
C_  RR*  ->  ( sup ( A ,  RR* ,  <  )  = +oo  <->  A. x  e.  RR  -.  A. y  e.  A  y  <_  x ) )
24 ralnex 2889 . . . . . . . 8  |-  ( A. x  e.  RR  -.  A. y  e.  A  y  <_  x  <->  -.  E. x  e.  RR  A. y  e.  A  y  <_  x
)
2523, 24syl6bb 261 . . . . . . 7  |-  ( A 
C_  RR*  ->  ( sup ( A ,  RR* ,  <  )  = +oo  <->  -.  E. x  e.  RR  A. y  e.  A  y  <_  x
) )
2625necon2abid 2697 . . . . . 6  |-  ( A 
C_  RR*  ->  ( E. x  e.  RR  A. y  e.  A  y  <_  x  <->  sup ( A ,  RR* ,  <  )  =/= +oo ) )
2710, 26sylibrd 234 . . . . 5  |-  ( A 
C_  RR*  ->  ( sup ( A ,  RR* ,  <  )  < +oo  ->  E. x  e.  RR  A. y  e.  A  y  <_  x
) )
2827imp 429 . . . 4  |-  ( ( A  C_  RR*  /\  sup ( A ,  RR* ,  <  )  < +oo )  ->  E. x  e.  RR  A. y  e.  A  y  <_  x
)
293, 28sylan 471 . . 3  |-  ( ( A  C_  RR  /\  sup ( A ,  RR* ,  <  )  < +oo )  ->  E. x  e.  RR  A. y  e.  A  y  <_  x
)
30293adant2 1016 . 2  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  sup ( A ,  RR* ,  <  )  < +oo )  ->  E. x  e.  RR  A. y  e.  A  y  <_  x
)
31 supxrre 11530 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR* ,  <  )  =  sup ( A ,  RR ,  <  ) )
32 suprcl 10510 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR ,  <  )  e.  RR )
3331, 32eqeltrd 2531 . 2  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR* ,  <  )  e.  RR )
3430, 33syld3an3 1274 1  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  sup ( A ,  RR* ,  <  )  < +oo )  ->  sup ( A ,  RR* ,  <  )  e.  RR )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 974    = wceq 1383    e. wcel 1804    =/= wne 2638   A.wral 2793   E.wrex 2794    C_ wss 3461   (/)c0 3770   class class class wbr 4437   supcsup 7902   RRcr 9494   +oocpnf 9628   RR*cxr 9630    < clt 9631    <_ cle 9632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-er 7313  df-en 7519  df-dom 7520  df-sdom 7521  df-sup 7903  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813
This theorem is referenced by:  supxrgtmnf  11532  ovolunlem1  21886  uniioombllem1  21968
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