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Theorem mnflt 11218
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 2454 . . . 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 11208 . . 3  |- -oo  e.  RR*
6 rexr 9543 . . 3  |-  ( A  e.  RR  ->  A  e.  RR* )
7 ltxr 11209 . . 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 1370    e. wcel 1758   class class class wbr 4403   RRcr 9395    <RR cltrr 9400   +oocpnf 9529   -oocmnf 9530   RR*cxr 9531    < clt 9532
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-cnex 9452
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-xp 4957  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537
This theorem is referenced by:  mnflt0  11219  mnfltxr  11221  xrlttri  11230  xrlttr  11231  xrrebnd  11254  xrre3  11257  qbtwnxr  11284  xltnegi  11300  xrsupsslem  11383  xrub  11388  supxrre  11404  infmxrre  11412  elico2  11473  elicc2  11474  ioomax  11484  elioomnf  11504  difreicc  11537  caucvgrlem  13271  icopnfcld  20482  iocmnfcld  20483  tgioo  20508  xrtgioo  20518  reconnlem1  20538  reconnlem2  20539  bndth  20665  ovoliunlem1  21120  ovoliun  21123  ioombl1lem2  21176  mbfmax  21263  ismbf3d  21268  itg2seq  21356  dvferm1lem  21592  dvferm2lem  21594  degltlem1  21679  ply1divex  21744  dvdsq1p  21768  ellogdm  22220  logdmnrp  22222  atans2  22462  dya2iocbrsiga  26854  dya2icobrsiga  26855  orvclteel  27019  itg2addnclem  28611  asindmre  28647  dvasin  28648  dvacos  28649  areacirclem5  28656  rfcnpre4  29924
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