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Theorem xrnepnf 11204
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 11200 . . . 4  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
3 df-3or 966 . . . 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 2646 . . 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 9535 . . . . 5  |-  ( A  e.  RR  ->  A  =/= +oo )
9 mnfnepnf 11202 . . . . . 6  |- -oo  =/= +oo
10 neeq1 2729 . . . . . 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 2651 . . 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 964    = wceq 1370    e. wcel 1758    =/= wne 2644   RRcr 9385   +oocpnf 9519   -oocmnf 9520   RR*cxr 9521
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-pw 3963  df-sn 3979  df-pr 3981  df-uni 4193  df-pnf 9524  df-mnf 9525  df-xr 9526
This theorem is referenced by:  xaddnepnf  11309  xlt2addrd  26195
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