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Theorem mnflt 11329
Description: Minus infinity is less than any (finite) real. (Contributed by NM, 14-Oct-2005.)
Assertion
Ref Expression
mnflt  |-  ( A  e.  RR  -> -oo  <  A )

Proof of Theorem mnflt
StepHypRef Expression
1 eqid 2467 . . . 4  |- -oo  = -oo
2 olc 384 . . . 4  |-  ( ( -oo  = -oo  /\  A  e.  RR )  ->  ( ( -oo  e.  RR  /\  A  = +oo )  \/  ( -oo  = -oo  /\  A  e.  RR ) ) )
31, 2mpan 670 . . 3  |-  ( A  e.  RR  ->  (
( -oo  e.  RR  /\  A  = +oo )  \/  ( -oo  = -oo  /\  A  e.  RR ) ) )
43olcd 393 . 2  |-  ( A  e.  RR  ->  (
( ( ( -oo  e.  RR  /\  A  e.  RR )  /\ -oo  <RR  A )  \/  ( -oo  = -oo  /\  A  = +oo ) )  \/  ( ( -oo  e.  RR  /\  A  = +oo )  \/  ( -oo  = -oo  /\  A  e.  RR ) ) ) )
5 mnfxr 11319 . . 3  |- -oo  e.  RR*
6 rexr 9635 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
7 ltxr 11320 . . 3  |-  ( ( -oo  e.  RR*  /\  A  e.  RR* )  ->  ( -oo  <  A  <->  ( (
( ( -oo  e.  RR  /\  A  e.  RR )  /\ -oo  <RR  A )  \/  ( -oo  = -oo  /\  A  = +oo ) )  \/  (
( -oo  e.  RR  /\  A  = +oo )  \/  ( -oo  = -oo  /\  A  e.  RR ) ) ) ) )
85, 6, 7sylancr 663 . 2  |-  ( A  e.  RR  ->  ( -oo  <  A  <->  ( (
( ( -oo  e.  RR  /\  A  e.  RR )  /\ -oo  <RR  A )  \/  ( -oo  = -oo  /\  A  = +oo ) )  \/  (
( -oo  e.  RR  /\  A  = +oo )  \/  ( -oo  = -oo  /\  A  e.  RR ) ) ) ) )
94, 8mpbird 232 1  |-  ( A  e.  RR  -> -oo  <  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767   class class class wbr 4447   RRcr 9487    <RR cltrr 9492   +oocpnf 9621   -oocmnf 9622   RR*cxr 9623    < clt 9624
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 6574  ax-cnex 9544
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  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-xp 5005  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629
This theorem is referenced by:  mnflt0  11330  mnfltxr  11332  xrlttri  11341  xrlttr  11342  xrrebnd  11365  xrre3  11368  qbtwnxr  11395  xltnegi  11411  xrsupsslem  11494  xrub  11499  supxrre  11515  infmxrre  11523  elico2  11584  elicc2  11585  ioomax  11595  elioomnf  11615  difreicc  11648  caucvgrlem  13451  icopnfcld  21007  iocmnfcld  21008  tgioo  21033  xrtgioo  21043  reconnlem1  21063  reconnlem2  21064  bndth  21190  ovoliunlem1  21645  ovoliun  21648  ioombl1lem2  21701  mbfmax  21788  ismbf3d  21793  itg2seq  21881  dvferm1lem  22117  dvferm2lem  22119  degltlem1  22204  ply1divex  22269  dvdsq1p  22293  ellogdm  22745  logdmnrp  22747  atans2  22987  dya2iocbrsiga  27883  dya2icobrsiga  27884  orvclteel  28048  itg2addnclem  29641  asindmre  29677  dvasin  29678  dvacos  29679  areacirclem5  29686  rfcnpre4  30987  mnfltd  31068  limciccioolb  31163  icccncfext  31226  cncfiooicclem1  31232  fourierdlem32  31439  fourierdlem87  31494  fourierdlem113  31520  fouriersw  31532
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