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Theorem supxrbnd 11519
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 9636 . . . . 5  |-  RR  C_  RR*
2 sstr 3512 . . . . 5  |-  ( ( A  C_  RR  /\  RR  C_ 
RR* )  ->  A  C_ 
RR* )
31, 2mpan2 671 . . . 4  |-  ( A 
C_  RR  ->  A  C_  RR* )
4 supxrcl 11505 . . . . . . 7  |-  ( A 
C_  RR*  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
5 pnfxr 11320 . . . . . . . . . 10  |- +oo  e.  RR*
6 xrltne 11365 . . . . . . . . . 10  |-  ( ( sup ( A ,  RR* ,  <  )  e. 
RR*  /\ +oo  e.  RR*  /\ 
sup ( A ,  RR* ,  <  )  < +oo )  -> +oo  =/=  sup ( A ,  RR* ,  <  ) )
75, 6mp3an2 1312 . . . . . . . . 9  |-  ( ( sup ( A ,  RR* ,  <  )  e. 
RR*  /\  sup ( A ,  RR* ,  <  )  < +oo )  -> +oo  =/=  sup ( A ,  RR* ,  <  ) )
87necomd 2738 . . . . . . . 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 11511 . . . . . . . . 9  |-  ( A 
C_  RR*  ->  ( A. x  e.  RR  E. y  e.  A  x  <  y  <->  sup ( A ,  RR* ,  <  )  = +oo ) )
12 ssel2 3499 . . . . . . . . . . . . . 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 9638 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  x  e.  RR* )
1514ad2antlr 726 . . . . . . . . . . . . 13  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  y  e.  A )  ->  x  e.  RR* )
16 xrlenlt 9651 . . . . . . . . . . . . . 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 2970 . . . . . . . . . . 11  |-  ( ( A  C_  RR*  /\  x  e.  RR )  ->  ( E. y  e.  A  x  <  y  <->  E. y  e.  A  -.  y  <_  x ) )
20 rexnal 2912 . . . . . . . . . . 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 2900 . . . . . . . . 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 2910 . . . . . . . 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 2721 . . . . . 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 1015 . 2  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  sup ( A ,  RR* ,  <  )  < +oo )  ->  E. x  e.  RR  A. y  e.  A  y  <_  x
)
31 supxrre 11518 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR* ,  <  )  =  sup ( A ,  RR ,  <  ) )
32 suprcl 10502 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR ,  <  )  e.  RR )
3331, 32eqeltrd 2555 . 2  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR* ,  <  )  e.  RR )
3430, 33syld3an3 1273 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   A.wral 2814   E.wrex 2815    C_ wss 3476   (/)c0 3785   class class class wbr 4447   supcsup 7899   RRcr 9490   +oocpnf 9624   RR*cxr 9626    < clt 9627    <_ cle 9628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568  ax-pre-sup 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-sup 7900  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807
This theorem is referenced by:  supxrgtmnf  11520  ovolunlem1  21659  uniioombllem1  21741
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