MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  supxrre2 Structured version   Unicode version

Theorem supxrre2 10910
Description: The supremum of a nonempty set of reals is real iff it is not plus infinity. (Contributed by NM, 5-Feb-2006.)
Assertion
Ref Expression
supxrre2  |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  ( sup ( A ,  RR* ,  <  )  e.  RR  <->  sup ( A ,  RR* ,  <  )  =/=  +oo ) )

Proof of Theorem supxrre2
StepHypRef Expression
1 supxrre1 10909 . 2  |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  ( sup ( A ,  RR* ,  <  )  e.  RR  <->  sup ( A ,  RR* ,  <  )  <  +oo ) )
2 ressxr 9129 . . . . . 6  |-  RR  C_  RR*
3 sstr 3356 . . . . . 6  |-  ( ( A  C_  RR  /\  RR  C_ 
RR* )  ->  A  C_ 
RR* )
42, 3mpan2 653 . . . . 5  |-  ( A 
C_  RR  ->  A  C_  RR* )
5 supxrcl 10893 . . . . 5  |-  ( A 
C_  RR*  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
6 nltpnft 10754 . . . . 5  |-  ( sup ( A ,  RR* ,  <  )  e.  RR*  ->  ( sup ( A ,  RR* ,  <  )  =  +oo  <->  -.  sup ( A ,  RR* ,  <  )  <  +oo ) )
74, 5, 63syl 19 . . . 4  |-  ( A 
C_  RR  ->  ( sup ( A ,  RR* ,  <  )  =  +oo  <->  -.  sup ( A ,  RR* ,  <  )  <  +oo ) )
87necon2abid 2661 . . 3  |-  ( A 
C_  RR  ->  ( sup ( A ,  RR* ,  <  )  <  +oo  <->  sup ( A ,  RR* ,  <  )  =/=  +oo ) )
98adantr 452 . 2  |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  ( sup ( A ,  RR* ,  <  )  <  +oo  <->  sup ( A ,  RR* ,  <  )  =/=  +oo ) )
101, 9bitrd 245 1  |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  ( sup ( A ,  RR* ,  <  )  e.  RR  <->  sup ( A ,  RR* ,  <  )  =/=  +oo ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599    C_ wss 3320   (/)c0 3628   class class class wbr 4212   supcsup 7445   RRcr 8989    +oocpnf 9117   RR*cxr 9119    < clt 9120
This theorem is referenced by:  ovollb2  19385  nmorepnf  22269  nmoprepnf  23370  nmfnrepnf  23383
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 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294
  Copyright terms: Public domain W3C validator