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Theorem supxrgtmnf 11533
Description: The supremum of a nonempty set of reals is greater than minus infinity. (Contributed by NM, 2-Feb-2006.)
Assertion
Ref Expression
supxrgtmnf  |-  ( ( A  C_  RR  /\  A  =/=  (/) )  -> -oo  <  sup ( A ,  RR* ,  <  ) )

Proof of Theorem supxrgtmnf
StepHypRef Expression
1 supxrbnd 11532 . . . . . 6  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  sup ( A ,  RR* ,  <  )  < +oo )  ->  sup ( A ,  RR* ,  <  )  e.  RR )
213expia 1198 . . . . 5  |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  ( sup ( A ,  RR* ,  <  )  < +oo  ->  sup ( A ,  RR* ,  <  )  e.  RR ) )
32con3d 133 . . . 4  |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  ( -.  sup ( A ,  RR* ,  <  )  e.  RR  ->  -.  sup ( A ,  RR* ,  <  )  < +oo ) )
4 ressxr 9649 . . . . . . . 8  |-  RR  C_  RR*
5 sstr 3517 . . . . . . . 8  |-  ( ( A  C_  RR  /\  RR  C_ 
RR* )  ->  A  C_ 
RR* )
64, 5mpan2 671 . . . . . . 7  |-  ( A 
C_  RR  ->  A  C_  RR* )
7 supxrcl 11518 . . . . . . 7  |-  ( A 
C_  RR*  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
86, 7syl 16 . . . . . 6  |-  ( A 
C_  RR  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
98adantr 465 . . . . 5  |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
10 nltpnft 11379 . . . . 5  |-  ( sup ( A ,  RR* ,  <  )  e.  RR*  ->  ( sup ( A ,  RR* ,  <  )  = +oo  <->  -.  sup ( A ,  RR* ,  <  )  < +oo ) )
119, 10syl 16 . . . 4  |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  ( sup ( A ,  RR* ,  <  )  = +oo  <->  -.  sup ( A ,  RR* ,  <  )  < +oo ) )
123, 11sylibrd 234 . . 3  |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  ( -.  sup ( A ,  RR* ,  <  )  e.  RR  ->  sup ( A ,  RR* ,  <  )  = +oo ) )
1312orrd 378 . 2  |-  ( ( A  C_  RR  /\  A  =/=  (/) )  ->  ( sup ( A ,  RR* ,  <  )  e.  RR  \/  sup ( A ,  RR* ,  <  )  = +oo ) )
14 mnfltxr 11348 . 2  |-  ( ( sup ( A ,  RR* ,  <  )  e.  RR  \/  sup ( A ,  RR* ,  <  )  = +oo )  -> -oo  <  sup ( A ,  RR* ,  <  ) )
1513, 14syl 16 1  |-  ( ( A  C_  RR  /\  A  =/=  (/) )  -> -oo  <  sup ( A ,  RR* ,  <  ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662    C_ wss 3481   (/)c0 3790   class class class wbr 4453   supcsup 7912   RRcr 9503   +oocpnf 9637   -oocmnf 9638   RR*cxr 9639    < clt 9640
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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820
This theorem is referenced by:  supxrre1  11534  ovolunlem1a  21775
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