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Theorem xrnemnf 11200
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 11197 . . . 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 ) )
42, 3bitri 249 . . 3  |-  ( A  e.  RR*  <->  ( ( A  e.  RR  \/  A  = +oo )  \/  A  = -oo ) )
5 df-ne 2646 . . 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 9533 . . . . 5  |-  ( A  e.  RR  ->  A  =/= -oo )
8 pnfnemnf 11198 . . . . . 6  |- +oo  =/= -oo
9 neeq1 2729 . . . . . 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 2651 . . 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 964    = wceq 1370    e. wcel 1758    =/= wne 2644   RRcr 9382   +oocpnf 9516   -oocmnf 9517   RR*cxr 9518
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 4511  ax-nul 4519  ax-pow 4568  ax-pr 4629  ax-un 6472  ax-cnex 9439  ax-resscn 9440
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-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-csb 3387  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-pw 3960  df-sn 3976  df-pr 3978  df-op 3982  df-uni 4190  df-br 4391  df-opab 4449  df-mpt 4450  df-id 4734  df-xp 4944  df-rel 4945  df-cnv 4946  df-co 4947  df-dm 4948  df-rn 4949  df-res 4950  df-ima 4951  df-iota 5479  df-fun 5518  df-fn 5519  df-f 5520  df-f1 5521  df-fo 5522  df-f1o 5523  df-fv 5524  df-er 7201  df-en 7411  df-dom 7412  df-sdom 7413  df-pnf 9521  df-mnf 9522  df-xr 9523
This theorem is referenced by:  xaddnemnf  11305  xaddass  11313  xlesubadd  11327  xblss2ps  20092  xblss2  20093  nmoix  20424  nmoleub  20426  blcvx  20491  xrge0tsms  20527  metdstri  20543  nmoleub2lem  20785  xrge0nre  26287  xrge0tsmsd  26387
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