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Theorem xrnepnf 10652
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 10649 . . . 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 2553 . . 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 9066 . . . . 5  |-  ( A  e.  RR  ->  A  =/=  +oo )
9 pnfnemnf 10650 . . . . . . 7  |-  +oo  =/=  -oo
109necomi 2633 . . . . . 6  |-  -oo  =/=  +oo
11 neeq1 2559 . . . . . 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 2567 . . 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 1649    e. wcel 1717    =/= wne 2551   RRcr 8923    +oocpnf 9051    -oocmnf 9052   RR*cxr 9053
This theorem is referenced by:  xaddnepnf  10754  xlt2addrd  23961
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-pw 3745  df-sn 3764  df-pr 3765  df-uni 3959  df-pnf 9056  df-mnf 9057  df-xr 9058
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