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Theorem xrnepnf 10711
Description: An extended real other than plus infinity is real or negative infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xrnepnf  |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  <->  ( A  e.  RR  \/  A  = 
-oo ) )

Proof of Theorem xrnepnf
StepHypRef Expression
1 pm5.61 694 . 2  |-  ( ( ( ( A  e.  RR  \/  A  = 
-oo )  \/  A  =  +oo )  /\  -.  A  =  +oo )  <->  ( ( A  e.  RR  \/  A  =  -oo )  /\  -.  A  =  +oo ) )
2 elxr 10708 . . . 4  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = 
+oo  \/  A  =  -oo ) )
3 df-3or 937 . . . 4  |-  ( ( A  e.  RR  \/  A  =  +oo  \/  A  =  -oo )  <->  ( ( A  e.  RR  \/  A  =  +oo )  \/  A  =  -oo )
)
4 or32 514 . . . 4  |-  ( ( ( A  e.  RR  \/  A  =  +oo )  \/  A  =  -oo )  <->  ( ( A  e.  RR  \/  A  =  -oo )  \/  A  =  +oo ) )
52, 3, 43bitri 263 . . 3  |-  ( A  e.  RR*  <->  ( ( A  e.  RR  \/  A  =  -oo )  \/  A  =  +oo ) )
6 df-ne 2600 . . 3  |-  ( A  =/=  +oo  <->  -.  A  =  +oo )
75, 6anbi12i 679 . 2  |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  <->  ( (
( A  e.  RR  \/  A  =  -oo )  \/  A  =  +oo )  /\  -.  A  =  +oo ) )
8 renepnf 9124 . . . . 5  |-  ( A  e.  RR  ->  A  =/=  +oo )
9 pnfnemnf 10709 . . . . . . 7  |-  +oo  =/=  -oo
109necomi 2680 . . . . . 6  |-  -oo  =/=  +oo
11 neeq1 2606 . . . . . 6  |-  ( A  =  -oo  ->  ( A  =/=  +oo  <->  -oo  =/=  +oo )
)
1210, 11mpbiri 225 . . . . 5  |-  ( A  =  -oo  ->  A  =/=  +oo )
138, 12jaoi 369 . . . 4  |-  ( ( A  e.  RR  \/  A  =  -oo )  ->  A  =/=  +oo )
1413neneqd 2614 . . 3  |-  ( ( A  e.  RR  \/  A  =  -oo )  ->  -.  A  =  +oo )
1514pm4.71i 614 . 2  |-  ( ( A  e.  RR  \/  A  =  -oo )  <->  ( ( A  e.  RR  \/  A  =  -oo )  /\  -.  A  =  +oo ) )
161, 7, 153bitr4i 269 1  |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  <->  ( A  e.  RR  \/  A  = 
-oo ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    \/ wo 358    /\ wa 359    \/ w3o 935    = wceq 1652    e. wcel 1725    =/= wne 2598   RRcr 8981    +oocpnf 9109    -oocmnf 9110   RR*cxr 9111
This theorem is referenced by:  xaddnepnf  10813  xlt2addrd  24116
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-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039
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-ne 2600  df-nel 2601  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  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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