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Theorem elxr 8694
Description: Membership in the set of extended reals. (Contributed by NM, 14-Oct-2005.)
Assertion
Ref Expression
elxr  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )

Proof of Theorem elxr
StepHypRef Expression
1 df-xr 7062 . . 3  |-  RR*  =  ( RR  u.  { +oo , -oo } )
21eleq2i 2104 . 2  |-  ( A  e.  RR*  <->  A  e.  ( RR  u.  { +oo , -oo } ) )
3 elun 3084 . 2  |-  ( A  e.  ( RR  u.  { +oo , -oo }
)  <->  ( A  e.  RR  \/  A  e. 
{ +oo , -oo }
) )
4 pnfex 8691 . . . . 5  |- +oo  e.  _V
5 mnfxr 8692 . . . . . 6  |- -oo  e.  RR*
65elexi 2567 . . . . 5  |- -oo  e.  _V
74, 6elpr2 3397 . . . 4  |-  ( A  e.  { +oo , -oo }  <->  ( A  = +oo  \/  A  = -oo ) )
87orbi2i 679 . . 3  |-  ( ( A  e.  RR  \/  A  e.  { +oo , -oo } )  <->  ( A  e.  RR  \/  ( A  = +oo  \/  A  = -oo ) ) )
9 3orass 888 . . 3  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  <->  ( A  e.  RR  \/  ( A  = +oo  \/  A  = -oo ) ) )
108, 9bitr4i 176 . 2  |-  ( ( A  e.  RR  \/  A  e.  { +oo , -oo } )  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
112, 3, 103bitri 195 1  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 98    \/ wo 629    \/ w3o 884    = wceq 1243    e. wcel 1393    u. cun 2915   {cpr 3376   RRcr 6886   +oocpnf 7055   -oocmnf 7056   RR*cxr 7057
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 6973
This theorem depends on definitions:  df-bi 110  df-3or 886  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 7060  df-mnf 7061  df-xr 7062
This theorem is referenced by:  xrnemnf  8697  xrnepnf  8698  xrltnr  8699  xrltnsym  8712  xrlttr  8714  xrltso  8715  xrlttri3  8716  nltpnft  8728  ngtmnft  8729  xrrebnd  8730  xnegcl  8743  xnegneg  8744  xltnegi  8746
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