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Theorem mnfxr 8694
Description: Minus infinity belongs to the set of extended reals. (Contributed by NM, 13-Oct-2005.) (Proof shortened by Anthony Hart, 29-Aug-2011.) (Proof shortened by Andrew Salmon, 19-Nov-2011.)
Assertion
Ref Expression
mnfxr  |- -oo  e.  RR*

Proof of Theorem mnfxr
StepHypRef Expression
1 df-mnf 7063 . . . . 5  |- -oo  =  ~P +oo
2 pnfex 8693 . . . . . 6  |- +oo  e.  _V
32pwex 3932 . . . . 5  |-  ~P +oo  e.  _V
41, 3eqeltri 2110 . . . 4  |- -oo  e.  _V
54prid2 3477 . . 3  |- -oo  e.  { +oo , -oo }
6 elun2 3111 . . 3  |-  ( -oo  e.  { +oo , -oo }  -> -oo  e.  ( RR  u.  { +oo , -oo } ) )
75, 6ax-mp 7 . 2  |- -oo  e.  ( RR  u.  { +oo , -oo } )
8 df-xr 7064 . 2  |-  RR*  =  ( RR  u.  { +oo , -oo } )
97, 8eleqtrri 2113 1  |- -oo  e.  RR*
Colors of variables: wff set class
Syntax hints:    e. wcel 1393   _Vcvv 2557    u. cun 2915   ~Pcpw 3359   {cpr 3376   RRcr 6888   +oocpnf 7057   -oocmnf 7058   RR*cxr 7059
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-un 4170  ax-cnex 6975
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-pnf 7062  df-mnf 7063  df-xr 7064
This theorem is referenced by:  elxr  8696  xrltnr  8701  mnflt  8704  mnfltpnf  8706  nltmnf  8709  mnfle  8713  xrltnsym  8714  xrlttri3  8718  ngtmnft  8731  xrrebnd  8732  xrre2  8734  xrre3  8735  ge0gtmnf  8736  xnegcl  8745  xltnegi  8748  xrex  8756  elioc2  8805  elico2  8806  elicc2  8807  ioomax  8817  iccmax  8818  elioomnf  8837  unirnioo  8842
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