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Theorem supxrbnd 10647
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 8876 . . . . 5  |-  RR  C_  RR*
2 sstr 3187 . . . . 5  |-  ( ( A  C_  RR  /\  RR  C_ 
RR* )  ->  A  C_ 
RR* )
31, 2mpan2 652 . . . 4  |-  ( A 
C_  RR  ->  A  C_  RR* )
4 supxrcl 10633 . . . . . . 7  |-  ( A 
C_  RR*  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
5 pnfxr 10455 . . . . . . . . . 10  |-  +oo  e.  RR*
6 xrltne 10494 . . . . . . . . . 10  |-  ( ( sup ( A ,  RR* ,  <  )  e. 
RR*  /\  +oo  e.  RR*  /\ 
sup ( A ,  RR* ,  <  )  <  +oo )  ->  +oo  =/=  sup ( A ,  RR* ,  <  ) )
75, 6mp3an2 1265 . . . . . . . . 9  |-  ( ( sup ( A ,  RR* ,  <  )  e. 
RR*  /\  sup ( A ,  RR* ,  <  )  <  +oo )  ->  +oo  =/=  sup ( A ,  RR* ,  <  ) )
87necomd 2529 . . . . . . . 8  |-  ( ( sup ( A ,  RR* ,  <  )  e. 
RR*  /\  sup ( A ,  RR* ,  <  )  <  +oo )  ->  sup ( A ,  RR* ,  <  )  =/=  +oo )
98ex 423 . . . . . . 7  |-  ( sup ( A ,  RR* ,  <  )  e.  RR*  ->  ( sup ( A ,  RR* ,  <  )  <  +oo  ->  sup ( A ,  RR* ,  <  )  =/=  +oo ) )
104, 9syl 15 . . . . . 6  |-  ( A 
C_  RR*  ->  ( sup ( A ,  RR* ,  <  )  <  +oo  ->  sup ( A ,  RR* ,  <  )  =/=  +oo ) )
11 supxrunb2 10639 . . . . . . . . 9  |-  ( A 
C_  RR*  ->  ( A. x  e.  RR  E. y  e.  A  x  <  y  <->  sup ( A ,  RR* ,  <  )  =  +oo ) )
12 ssel2 3175 . . . . . . . . . . . . . 14  |-  ( ( A  C_  RR*  /\  y  e.  A )  ->  y  e.  RR* )
1312adantlr 695 . . . . . . . . . . . . 13  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  y  e.  A )  ->  y  e.  RR* )
14 rexr 8877 . . . . . . . . . . . . . 14  |-  ( x  e.  RR  ->  x  e.  RR* )
1514ad2antlr 707 . . . . . . . . . . . . 13  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  y  e.  A )  ->  x  e.  RR* )
16 xrlenlt 8890 . . . . . . . . . . . . . 14  |-  ( ( y  e.  RR*  /\  x  e.  RR* )  ->  (
y  <_  x  <->  -.  x  <  y ) )
1716con2bid 319 . . . . . . . . . . . . 13  |-  ( ( y  e.  RR*  /\  x  e.  RR* )  ->  (
x  <  y  <->  -.  y  <_  x ) )
1813, 15, 17syl2anc 642 . . . . . . . . . . . 12  |-  ( ( ( A  C_  RR*  /\  x  e.  RR )  /\  y  e.  A )  ->  (
x  <  y  <->  -.  y  <_  x ) )
1918rexbidva 2560 . . . . . . . . . . 11  |-  ( ( A  C_  RR*  /\  x  e.  RR )  ->  ( E. y  e.  A  x  <  y  <->  E. y  e.  A  -.  y  <_  x ) )
20 rexnal 2554 . . . . . . . . . . 11  |-  ( E. y  e.  A  -.  y  <_  x  <->  -.  A. y  e.  A  y  <_  x )
2119, 20syl6bb 252 . . . . . . . . . 10  |-  ( ( A  C_  RR*  /\  x  e.  RR )  ->  ( E. y  e.  A  x  <  y  <->  -.  A. y  e.  A  y  <_  x ) )
2221ralbidva 2559 . . . . . . . . 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 246 . . . . . . . 8  |-  ( A 
C_  RR*  ->  ( sup ( A ,  RR* ,  <  )  =  +oo  <->  A. x  e.  RR  -.  A. y  e.  A  y  <_  x ) )
24 ralnex 2553 . . . . . . . 8  |-  ( A. x  e.  RR  -.  A. y  e.  A  y  <_  x  <->  -.  E. x  e.  RR  A. y  e.  A  y  <_  x
)
2523, 24syl6bb 252 . . . . . . 7  |-  ( A 
C_  RR*  ->  ( sup ( A ,  RR* ,  <  )  =  +oo  <->  -.  E. x  e.  RR  A. y  e.  A  y  <_  x
) )
2625necon2abid 2503 . . . . . 6  |-  ( A 
C_  RR*  ->  ( E. x  e.  RR  A. y  e.  A  y  <_  x  <->  sup ( A ,  RR* ,  <  )  =/=  +oo ) )
2710, 26sylibrd 225 . . . . 5  |-  ( A 
C_  RR*  ->  ( sup ( A ,  RR* ,  <  )  <  +oo  ->  E. x  e.  RR  A. y  e.  A  y  <_  x
) )
2827imp 418 . . . 4  |-  ( ( A  C_  RR*  /\  sup ( A ,  RR* ,  <  )  <  +oo )  ->  E. x  e.  RR  A. y  e.  A  y  <_  x
)
293, 28sylan 457 . . 3  |-  ( ( A  C_  RR  /\  sup ( A ,  RR* ,  <  )  <  +oo )  ->  E. x  e.  RR  A. y  e.  A  y  <_  x
)
30293adant2 974 . 2  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  sup ( A ,  RR* ,  <  )  <  +oo )  ->  E. x  e.  RR  A. y  e.  A  y  <_  x
)
31 supxrre 10646 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR* ,  <  )  =  sup ( A ,  RR ,  <  ) )
32 suprcl 9714 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR ,  <  )  e.  RR )
3331, 32eqeltrd 2357 . 2  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <_  x
)  ->  sup ( A ,  RR* ,  <  )  e.  RR )
3430, 33syld3an3 1227 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 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543   E.wrex 2544    C_ wss 3152   (/)c0 3455   class class class wbr 4023   supcsup 7193   RRcr 8736    +oocpnf 8864   RR*cxr 8866    < clt 8867    <_ cle 8868
This theorem is referenced by:  supxrgtmnf  10648  ovolunlem1  18856  uniioombllem1  18936
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-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  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-riota 6304  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
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