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Theorem xrnepnf 11366
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 11362 . . . 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 2596 . . 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 9634 . . . . 5  |-  ( A  e.  RR  ->  A  =/= +oo )
9 mnfnepnf 11364 . . . . . 6  |- -oo  =/= +oo
10 neeq1 2658 . . . . . 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 2601 . . 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 1872    =/= wne 2594   RRcr 9484   +oocpnf 9618   -oocmnf 9619   RR*cxr 9620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2403  ax-sep 4484  ax-nul 4493  ax-pow 4540  ax-pr 4598  ax-un 6536  ax-cnex 9541  ax-resscn 9542
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 1658  df-nf 1662  df-sb 1791  df-clab 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-ne 2596  df-nel 2597  df-rex 2715  df-rab 2718  df-v 3019  df-dif 3377  df-un 3379  df-in 3381  df-ss 3388  df-nul 3700  df-pw 3921  df-sn 3937  df-pr 3939  df-uni 4158  df-pnf 9623  df-mnf 9624  df-xr 9625
This theorem is referenced by:  xaddnepnf  11474  xlt2addrd  28283  xrlexaddrp  37472
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