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Theorem elxr 10708
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 9116 . . 3  |-  RR*  =  ( RR  u.  {  +oo , 
-oo } )
21eleq2i 2499 . 2  |-  ( A  e.  RR*  <->  A  e.  ( RR  u.  {  +oo ,  -oo } ) )
3 elun 3480 . 2  |-  ( A  e.  ( RR  u.  { 
+oo ,  -oo } )  <-> 
( A  e.  RR  \/  A  e.  {  +oo , 
-oo } ) )
4 pnfxr 10705 . . . . . 6  |-  +oo  e.  RR*
54elexi 2957 . . . . 5  |-  +oo  e.  _V
6 mnfxr 10706 . . . . . 6  |-  -oo  e.  RR*
76elexi 2957 . . . . 5  |-  -oo  e.  _V
85, 7elpr2 3825 . . . 4  |-  ( A  e.  {  +oo ,  -oo }  <->  ( A  = 
+oo  \/  A  =  -oo ) )
98orbi2i 506 . . 3  |-  ( ( A  e.  RR  \/  A  e.  {  +oo ,  -oo } )  <->  ( A  e.  RR  \/  ( A  =  +oo  \/  A  =  -oo ) ) )
10 3orass 939 . . 3  |-  ( ( A  e.  RR  \/  A  =  +oo  \/  A  =  -oo )  <->  ( A  e.  RR  \/  ( A  =  +oo  \/  A  =  -oo ) ) )
119, 10bitr4i 244 . 2  |-  ( ( A  e.  RR  \/  A  e.  {  +oo ,  -oo } )  <->  ( A  e.  RR  \/  A  = 
+oo  \/  A  =  -oo ) )
122, 3, 113bitri 263 1  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = 
+oo  \/  A  =  -oo ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    \/ wo 358    \/ w3o 935    = wceq 1652    e. wcel 1725    u. cun 3310   {cpr 3807   RRcr 8981    +oocpnf 9109    -oocmnf 9110   RR*cxr 9111
This theorem is referenced by:  xrnemnf  10710  xrnepnf  10711  xrltnr  10712  xrltnsym  10722  xrlttri  10724  xrlttr  10725  xrrebnd  10748  qbtwnxr  10778  xnegcl  10791  xnegneg  10792  xltnegi  10794  xaddf  10802  xnegid  10814  xaddcom  10816  xaddid1  10817  xnegdi  10819  xleadd1a  10824  xlt2add  10831  xsubge0  10832  xmullem  10835  xmulid1  10850  xmulgt0  10854  xmulasslem3  10857  xlemul1a  10859  xadddilem  10865  xadddi2  10868  xrsupsslem  10877  xrinfmsslem  10878  xrub  10882  isxmet2d  18349  blssioo  18818  ioombl1  19448  ismbf2d  19525  itg2seq  19626  xaddeq0  24111
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 2416  ax-sep 4322  ax-pow 4369  ax-un 4693  ax-cnex 9038
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-rex 2703  df-v 2950  df-un 3317  df-in 3319  df-ss 3326  df-pw 3793  df-sn 3812  df-pr 3813  df-uni 4008  df-pnf 9114  df-mnf 9115  df-xr 9116
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