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Theorem xrnemnf 11340
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 pm5.61 712 . 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 ) )
42, 3bitri 249 . . 3  |-  ( A  e.  RR*  <->  ( ( A  e.  RR  \/  A  = +oo )  \/  A  = -oo ) )
5 df-ne 2664 . . 3  |-  ( A  =/= -oo  <->  -.  A  = -oo )
64, 5anbi12i 697 . 2  |-  ( ( A  e.  RR*  /\  A  =/= -oo )  <->  ( (
( A  e.  RR  \/  A  = +oo )  \/  A  = -oo )  /\  -.  A  = -oo ) )
7 renemnf 9654 . . . . 5  |-  ( A  e.  RR  ->  A  =/= -oo )
8 pnfnemnf 11338 . . . . . 6  |- +oo  =/= -oo
9 neeq1 2748 . . . . . 6  |-  ( A  = +oo  ->  ( A  =/= -oo  <-> +oo  =/= -oo )
)
108, 9mpbiri 233 . . . . 5  |-  ( A  = +oo  ->  A  =/= -oo )
117, 10jaoi 379 . . . 4  |-  ( ( A  e.  RR  \/  A  = +oo )  ->  A  =/= -oo )
1211neneqd 2669 . . 3  |-  ( ( A  e.  RR  \/  A  = +oo )  ->  -.  A  = -oo )
1312pm4.71i 632 . 2  |-  ( ( A  e.  RR  \/  A  = +oo )  <->  ( ( A  e.  RR  \/  A  = +oo )  /\  -.  A  = -oo ) )
141, 6, 133bitr4i 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-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644
This theorem is referenced by:  xaddnemnf  11445  xaddass  11453  xlesubadd  11467  xblss2ps  20772  xblss2  20773  nmoix  21104  nmoleub  21106  blcvx  21171  xrge0tsms  21207  metdstri  21223  nmoleub2lem  21465  xrge0nre  27504  xrge0tsmsd  27600
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