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Theorem xrnepnf 10463
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 pm5.61 693 . 2  |-  ( ( ( ( A  e.  RR  \/  A  = 
-oo )  \/  A  =  +oo )  /\  -.  A  =  +oo )  <->  ( ( A  e.  RR  \/  A  =  -oo )  /\  -.  A  =  +oo ) )
2 elxr 10460 . . . 4  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = 
+oo  \/  A  =  -oo ) )
3 df-3or 935 . . . 4  |-  ( ( A  e.  RR  \/  A  =  +oo  \/  A  =  -oo )  <->  ( ( A  e.  RR  \/  A  =  +oo )  \/  A  =  -oo )
)
4 or32 513 . . . 4  |-  ( ( ( A  e.  RR  \/  A  =  +oo )  \/  A  =  -oo )  <->  ( ( A  e.  RR  \/  A  =  -oo )  \/  A  =  +oo ) )
52, 3, 43bitri 262 . . 3  |-  ( A  e.  RR*  <->  ( ( A  e.  RR  \/  A  =  -oo )  \/  A  =  +oo ) )
6 df-ne 2450 . . 3  |-  ( A  =/=  +oo  <->  -.  A  =  +oo )
75, 6anbi12i 678 . 2  |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  <->  ( (
( A  e.  RR  \/  A  =  -oo )  \/  A  =  +oo )  /\  -.  A  =  +oo ) )
8 renepnf 8881 . . . . 5  |-  ( A  e.  RR  ->  A  =/=  +oo )
9 pnfnemnf 10461 . . . . . . 7  |-  +oo  =/=  -oo
109necomi 2530 . . . . . 6  |-  -oo  =/=  +oo
11 neeq1 2456 . . . . . 6  |-  ( A  =  -oo  ->  ( A  =/=  +oo  <->  -oo  =/=  +oo )
)
1210, 11mpbiri 224 . . . . 5  |-  ( A  =  -oo  ->  A  =/=  +oo )
138, 12jaoi 368 . . . 4  |-  ( ( A  e.  RR  \/  A  =  -oo )  ->  A  =/=  +oo )
1413neneqd 2464 . . 3  |-  ( ( A  e.  RR  \/  A  =  -oo )  ->  -.  A  =  +oo )
1514pm4.71i 613 . 2  |-  ( ( A  e.  RR  \/  A  =  -oo )  <->  ( ( A  e.  RR  \/  A  =  -oo )  /\  -.  A  =  +oo ) )
161, 7, 153bitr4i 268 1  |-  ( ( A  e.  RR*  /\  A  =/=  +oo )  <->  ( A  e.  RR  \/  A  = 
-oo ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 176    \/ wo 357    /\ wa 358    \/ w3o 933    = wceq 1625    e. wcel 1686    =/= wne 2448   RRcr 8738    +oocpnf 8866    -oocmnf 8867   RR*cxr 8868
This theorem is referenced by:  xaddnepnf  10564  xlt2addrd  23255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1535  ax-5 1546  ax-17 1605  ax-9 1637  ax-8 1645  ax-13 1688  ax-14 1690  ax-6 1705  ax-7 1710  ax-11 1717  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pow 4190  ax-pr 4216  ax-un 4514  ax-cnex 8795  ax-resscn 8796
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-tru 1310  df-ex 1531  df-nf 1534  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-pw 3629  df-sn 3648  df-pr 3649  df-uni 3830  df-pnf 8871  df-mnf 8872  df-xr 8873
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