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

Theorem icopnfsup 11246
Description: A set of upper reals is unbounded above. (Contributed by Mario Carneiro, 7-May-2016.)
Assertion
Ref Expression
icopnfsup  |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  ->  sup ( ( A [,)  +oo ) ,  RR* ,  <  )  =  +oo )

Proof of Theorem icopnfsup
Dummy variables  x  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simpl 444 . 2  |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  ->  A  e.  RR* )
2 pnfxr 10713 . . 3  |-  +oo  e.  RR*
32a1i 11 . 2  |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  ->  +oo  e.  RR* )
4 nltpnft 10754 . . . . . 6  |-  ( A  e.  RR*  ->  ( A  =  +oo  <->  -.  A  <  +oo ) )
54necon2abid 2661 . . . . 5  |-  ( A  e.  RR*  ->  ( A  <  +oo  <->  A  =/=  +oo )
)
65biimpar 472 . . . 4  |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  ->  A  <  +oo )
7 lbico1 10966 . . . 4  |-  ( ( A  e.  RR*  /\  +oo  e.  RR*  /\  A  <  +oo )  ->  A  e.  ( A [,)  +oo ) )
81, 3, 6, 7syl3anc 1184 . . 3  |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  ->  A  e.  ( A [,)  +oo ) )
9 ne0i 3634 . . 3  |-  ( A  e.  ( A [,)  +oo )  ->  ( A [,)  +oo )  =/=  (/) )
108, 9syl 16 . 2  |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  ->  ( A [,)  +oo )  =/=  (/) )
11 df-ico 10922 . . 3  |-  [,)  =  ( x  e.  RR* ,  y  e.  RR*  |->  { z  e.  RR*  |  (
x  <_  z  /\  z  <  y ) } )
12 idd 22 . . 3  |-  ( ( w  e.  RR*  /\  +oo  e.  RR* )  ->  (
w  <  +oo  ->  w  <  +oo ) )
13 xrltle 10742 . . 3  |-  ( ( w  e.  RR*  /\  +oo  e.  RR* )  ->  (
w  <  +oo  ->  w  <_  +oo ) )
14 xrltle 10742 . . 3  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <  w  ->  A  <_  w ) )
15 idd 22 . . 3  |-  ( ( A  e.  RR*  /\  w  e.  RR* )  ->  ( A  <_  w  ->  A  <_  w ) )
1611, 12, 13, 14, 15ixxub 10937 . 2  |-  ( ( A  e.  RR*  /\  +oo  e.  RR*  /\  ( A [,)  +oo )  =/=  (/) )  ->  sup ( ( A [,)  +oo ) ,  RR* ,  <  )  =  +oo )
171, 3, 10, 16syl3anc 1184 1  |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  ->  sup ( ( A [,)  +oo ) ,  RR* ,  <  )  =  +oo )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   (/)c0 3628   class class class wbr 4212  (class class class)co 6081   supcsup 7445    +oocpnf 9117   RR*cxr 9119    < clt 9120    <_ cle 9121   [,)cico 10918
This theorem is referenced by:  dvfsumrlimge0  19914  dvfsumrlim2  19916
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-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  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-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  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  df-div 9678  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-q 10575  df-ico 10922
  Copyright terms: Public domain W3C validator