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Theorem xrnemnf 9504
Description: An extended real other than minus infinity is real or positive infinite. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xrnemnf  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  <->  ( A  e.  RR  \/  A  = +oo ) )

Proof of Theorem xrnemnf
StepHypRef Expression
1 pm5.61 766 . 2  |-  ( ( ( ( A  e.  RR  \/  A  = +oo )  \/  A  = -oo )  /\  -.  A  = -oo )  <->  ( ( A  e.  RR  \/  A  = +oo )  /\  -.  A  = -oo ) )
2 elxr 9503 . . . 4  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
3 df-3or 946 . . . 4  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  <->  ( ( A  e.  RR  \/  A  = +oo )  \/  A  = -oo ) )
42, 3bitri 183 . . 3  |-  ( A  e.  RR*  <->  ( ( A  e.  RR  \/  A  = +oo )  \/  A  = -oo ) )
5 df-ne 2284 . . 3  |-  ( A  =/= -oo  <->  -.  A  = -oo )
64, 5anbi12i 453 . 2  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  <->  ( (
( A  e.  RR  \/  A  = +oo )  \/  A  = -oo )  /\  -.  A  = -oo ) )
7 renemnf 7778 . . . . 5  |-  ( A  e.  RR  ->  A  =/= -oo )
8 pnfnemnf 7784 . . . . . 6  |- +oo  =/= -oo
9 neeq1 2296 . . . . . 6  |-  ( A  = +oo  ->  ( A  =/= -oo  <-> +oo  =/= -oo )
)
108, 9mpbiri 167 . . . . 5  |-  ( A  = +oo  ->  A  =/= -oo )
117, 10jaoi 688 . . . 4  |-  ( ( A  e.  RR  \/  A  = +oo )  ->  A  =/= -oo )
1211neneqd 2304 . . 3  |-  ( ( A  e.  RR  \/  A  = +oo )  ->  -.  A  = -oo )
1312pm4.71i 386 . 2  |-  ( ( A  e.  RR  \/  A  = +oo )  <->  ( ( A  e.  RR  \/  A  = +oo )  /\  -.  A  = -oo ) )
141, 6, 133bitr4i 211 1  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  <->  ( A  e.  RR  \/  A  = +oo ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 103    <-> wb 104    \/ wo 680    \/ w3o 944    = wceq 1314    e. wcel 1463    =/= wne 2283   RRcr 7583   +oocpnf 7761   -oocmnf 7762   RR*cxr 7763
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-un 4323  ax-setind 4420  ax-cnex 7675  ax-resscn 7676
This theorem depends on definitions:  df-bi 116  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-uni 3705  df-pnf 7766  df-mnf 7767  df-xr 7768
This theorem is referenced by:  xaddf  9567  xaddval  9568  xaddnemnf  9580  xaddass  9592  xlesubadd  9606  xblss2ps  12468  xblss2  12469
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