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Theorem xrnemnf 8800
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 pm5.61 718 . 2  |-  ( ( ( ( A  e.  RR  \/  A  = +oo )  \/  A  = -oo )  /\  -.  A  = -oo )  <->  ( ( A  e.  RR  \/  A  = +oo )  /\  -.  A  = -oo ) )
2 elxr 8797 . . . 4  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
3 df-3or 897 . . . 4  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  <->  ( ( A  e.  RR  \/  A  = +oo )  \/  A  = -oo ) )
42, 3bitri 177 . . 3  |-  ( A  e.  RR*  <->  ( ( A  e.  RR  \/  A  = +oo )  \/  A  = -oo ) )
5 df-ne 2221 . . 3  |-  ( A  =/= -oo  <->  -.  A  = -oo )
64, 5anbi12i 441 . 2  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  <->  ( (
( A  e.  RR  \/  A  = +oo )  \/  A  = -oo )  /\  -.  A  = -oo ) )
7 renemnf 7133 . . . . 5  |-  ( A  e.  RR  ->  A  =/= -oo )
8 pnfnemnf 8798 . . . . . 6  |- +oo  =/= -oo
9 neeq1 2233 . . . . . 6  |-  ( A  = +oo  ->  ( A  =/= -oo  <-> +oo  =/= -oo )
)
108, 9mpbiri 161 . . . . 5  |-  ( A  = +oo  ->  A  =/= -oo )
117, 10jaoi 646 . . . 4  |-  ( ( A  e.  RR  \/  A  = +oo )  ->  A  =/= -oo )
1211neneqd 2241 . . 3  |-  ( ( A  e.  RR  \/  A  = +oo )  ->  -.  A  = -oo )
1312pm4.71i 377 . 2  |-  ( ( A  e.  RR  \/  A  = +oo )  <->  ( ( A  e.  RR  \/  A  = +oo )  /\  -.  A  = -oo ) )
141, 6, 133bitr4i 205 1  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  <->  ( A  e.  RR  \/  A  = +oo ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 101    <-> wb 102    \/ wo 639    \/ w3o 895    = wceq 1259    e. wcel 1409    =/= wne 2220   RRcr 6946   +oocpnf 7116   -oocmnf 7117   RR*cxr 7118
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-un 4198  ax-setind 4290  ax-cnex 7033  ax-resscn 7034
This theorem depends on definitions:  df-bi 114  df-3or 897  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-nel 2315  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-uni 3609  df-pnf 7121  df-mnf 7122  df-xr 7123
This theorem is referenced by: (None)
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