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Theorem xrnepnf 11420
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 pm5.61 717 . 2  |-  ( ( ( ( A  e.  RR  \/  A  = -oo )  \/  A  = +oo )  /\  -.  A  = +oo )  <->  ( ( A  e.  RR  \/  A  = -oo )  /\  -.  A  = +oo ) )
2 elxr 11416 . . . 4  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
3 df-3or 983 . . . 4  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  <->  ( ( A  e.  RR  \/  A  = +oo )  \/  A  = -oo ) )
4 or32 529 . . . 4  |-  ( ( ( A  e.  RR  \/  A  = +oo )  \/  A  = -oo )  <->  ( ( A  e.  RR  \/  A  = -oo )  \/  A  = +oo ) )
52, 3, 43bitri 274 . . 3  |-  ( A  e.  RR*  <->  ( ( A  e.  RR  \/  A  = -oo )  \/  A  = +oo ) )
6 df-ne 2627 . . 3  |-  ( A  =/= +oo  <->  -.  A  = +oo )
75, 6anbi12i 701 . 2  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  <->  ( (
( A  e.  RR  \/  A  = -oo )  \/  A  = +oo )  /\  -.  A  = +oo ) )
8 renepnf 9687 . . . . 5  |-  ( A  e.  RR  ->  A  =/= +oo )
9 mnfnepnf 11418 . . . . . 6  |- -oo  =/= +oo
10 neeq1 2712 . . . . . 6  |-  ( A  = -oo  ->  ( A  =/= +oo  <-> -oo  =/= +oo )
)
119, 10mpbiri 236 . . . . 5  |-  ( A  = -oo  ->  A  =/= +oo )
128, 11jaoi 380 . . . 4  |-  ( ( A  e.  RR  \/  A  = -oo )  ->  A  =/= +oo )
1312neneqd 2632 . . 3  |-  ( ( A  e.  RR  \/  A  = -oo )  ->  -.  A  = +oo )
1413pm4.71i 636 . 2  |-  ( ( A  e.  RR  \/  A  = -oo )  <->  ( ( A  e.  RR  \/  A  = -oo )  /\  -.  A  = +oo ) )
151, 7, 143bitr4i 280 1  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  <->  ( A  e.  RR  \/  A  = -oo ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    <-> wb 187    \/ wo 369    /\ wa 370    \/ w3o 981    = wceq 1437    e. wcel 1870    =/= wne 2625   RRcr 9537   +oocpnf 9671   -oocmnf 9672   RR*cxr 9673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-pw 3987  df-sn 4003  df-pr 4005  df-uni 4223  df-pnf 9676  df-mnf 9677  df-xr 9678
This theorem is referenced by:  xaddnepnf  11528  xlt2addrd  28179
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