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Theorem xrnepnf 9218
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 pm5.61 743 . 2  |-  ( ( ( ( A  e.  RR  \/  A  = -oo )  \/  A  = +oo )  /\  -.  A  = +oo )  <->  ( ( A  e.  RR  \/  A  = -oo )  /\  -.  A  = +oo ) )
2 elxr 9216 . . . 4  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
3 df-3or 925 . . . 4  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  <->  ( ( A  e.  RR  \/  A  = +oo )  \/  A  = -oo ) )
4 or32 722 . . . 4  |-  ( ( ( A  e.  RR  \/  A  = +oo )  \/  A  = -oo )  <->  ( ( A  e.  RR  \/  A  = -oo )  \/  A  = +oo ) )
52, 3, 43bitri 204 . . 3  |-  ( A  e.  RR*  <->  ( ( A  e.  RR  \/  A  = -oo )  \/  A  = +oo ) )
6 df-ne 2256 . . 3  |-  ( A  =/= +oo  <->  -.  A  = +oo )
75, 6anbi12i 448 . 2  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  <->  ( (
( A  e.  RR  \/  A  = -oo )  \/  A  = +oo )  /\  -.  A  = +oo ) )
8 renepnf 7514 . . . . 5  |-  ( A  e.  RR  ->  A  =/= +oo )
9 mnfnepnf 7522 . . . . . 6  |- -oo  =/= +oo
10 neeq1 2268 . . . . . 6  |-  ( A  = -oo  ->  ( A  =/= +oo  <-> -oo  =/= +oo )
)
119, 10mpbiri 166 . . . . 5  |-  ( A  = -oo  ->  A  =/= +oo )
128, 11jaoi 671 . . . 4  |-  ( ( A  e.  RR  \/  A  = -oo )  ->  A  =/= +oo )
1312neneqd 2276 . . 3  |-  ( ( A  e.  RR  \/  A  = -oo )  ->  -.  A  = +oo )
1413pm4.71i 383 . 2  |-  ( ( A  e.  RR  \/  A  = -oo )  <->  ( ( A  e.  RR  \/  A  = -oo )  /\  -.  A  = +oo ) )
151, 7, 143bitr4i 210 1  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  <->  ( A  e.  RR  \/  A  = -oo ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 102    <-> wb 103    \/ wo 664    \/ w3o 923    = wceq 1289    e. wcel 1438    =/= wne 2255   RRcr 7328   +oocpnf 7498   -oocmnf 7499   RR*cxr 7500
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-un 4251  ax-cnex 7415  ax-resscn 7416
This theorem depends on definitions:  df-bi 115  df-3or 925  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-rex 2365  df-rab 2368  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-uni 3649  df-pnf 7503  df-mnf 7504  df-xr 7505
This theorem is referenced by: (None)
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