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Theorem xrnepnf 11341
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 pm5.61 712 . 2  |-  ( ( ( ( A  e.  RR  \/  A  = -oo )  \/  A  = +oo )  /\  -.  A  = +oo )  <->  ( ( A  e.  RR  \/  A  = -oo )  /\  -.  A  = +oo ) )
2 elxr 11337 . . . 4  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
3 df-3or 974 . . . 4  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  <->  ( ( A  e.  RR  \/  A  = +oo )  \/  A  = -oo ) )
4 or32 527 . . . 4  |-  ( ( ( A  e.  RR  \/  A  = +oo )  \/  A  = -oo )  <->  ( ( A  e.  RR  \/  A  = -oo )  \/  A  = +oo ) )
52, 3, 43bitri 271 . . 3  |-  ( A  e.  RR*  <->  ( ( A  e.  RR  \/  A  = -oo )  \/  A  = +oo ) )
6 df-ne 2664 . . 3  |-  ( A  =/= +oo  <->  -.  A  = +oo )
75, 6anbi12i 697 . 2  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  <->  ( (
( A  e.  RR  \/  A  = -oo )  \/  A  = +oo )  /\  -.  A  = +oo ) )
8 renepnf 9653 . . . . 5  |-  ( A  e.  RR  ->  A  =/= +oo )
9 mnfnepnf 11339 . . . . . 6  |- -oo  =/= +oo
10 neeq1 2748 . . . . . 6  |-  ( A  = -oo  ->  ( A  =/= +oo  <-> -oo  =/= +oo )
)
119, 10mpbiri 233 . . . . 5  |-  ( A  = -oo  ->  A  =/= +oo )
128, 11jaoi 379 . . . 4  |-  ( ( A  e.  RR  \/  A  = -oo )  ->  A  =/= +oo )
1312neneqd 2669 . . 3  |-  ( ( A  e.  RR  \/  A  = -oo )  ->  -.  A  = +oo )
1413pm4.71i 632 . 2  |-  ( ( A  e.  RR  \/  A  = -oo )  <->  ( ( A  e.  RR  \/  A  = -oo )  /\  -.  A  = +oo ) )
151, 7, 143bitr4i 277 1  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  <->  ( A  e.  RR  \/  A  = -oo ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 184    \/ wo 368    /\ wa 369    \/ w3o 972    = wceq 1379    e. wcel 1767    =/= wne 2662   RRcr 9503   +oocpnf 9637   -oocmnf 9638   RR*cxr 9639
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-pw 4018  df-sn 4034  df-pr 4036  df-uni 4252  df-pnf 9642  df-mnf 9643  df-xr 9644
This theorem is referenced by:  xaddnepnf  11446  xlt2addrd  27392
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