MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  xrnemnf Structured version   Unicode version

Theorem xrnemnf 10710
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 pm5.61 694 . 2  |-  ( ( ( ( A  e.  RR  \/  A  = 
+oo )  \/  A  =  -oo )  /\  -.  A  =  -oo )  <->  ( ( A  e.  RR  \/  A  =  +oo )  /\  -.  A  =  -oo ) )
2 elxr 10708 . . . 4  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = 
+oo  \/  A  =  -oo ) )
3 df-3or 937 . . . 4  |-  ( ( A  e.  RR  \/  A  =  +oo  \/  A  =  -oo )  <->  ( ( A  e.  RR  \/  A  =  +oo )  \/  A  =  -oo )
)
42, 3bitri 241 . . 3  |-  ( A  e.  RR*  <->  ( ( A  e.  RR  \/  A  =  +oo )  \/  A  =  -oo ) )
5 df-ne 2600 . . 3  |-  ( A  =/=  -oo  <->  -.  A  =  -oo )
64, 5anbi12i 679 . 2  |-  ( ( A  e.  RR*  /\  A  =/=  -oo )  <->  ( (
( A  e.  RR  \/  A  =  +oo )  \/  A  =  -oo )  /\  -.  A  =  -oo ) )
7 renemnf 9125 . . . . 5  |-  ( A  e.  RR  ->  A  =/=  -oo )
8 pnfnemnf 10709 . . . . . 6  |-  +oo  =/=  -oo
9 neeq1 2606 . . . . . 6  |-  ( A  =  +oo  ->  ( A  =/=  -oo  <->  +oo  =/=  -oo )
)
108, 9mpbiri 225 . . . . 5  |-  ( A  =  +oo  ->  A  =/=  -oo )
117, 10jaoi 369 . . . 4  |-  ( ( A  e.  RR  \/  A  =  +oo )  ->  A  =/=  -oo )
1211neneqd 2614 . . 3  |-  ( ( A  e.  RR  \/  A  =  +oo )  ->  -.  A  =  -oo )
1312pm4.71i 614 . 2  |-  ( ( A  e.  RR  \/  A  =  +oo )  <->  ( ( A  e.  RR  \/  A  =  +oo )  /\  -.  A  =  -oo ) )
141, 6, 133bitr4i 269 1  |-  ( ( A  e.  RR*  /\  A  =/=  -oo )  <->  ( A  e.  RR  \/  A  = 
+oo ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 177    \/ wo 358    /\ wa 359    \/ w3o 935    = wceq 1652    e. wcel 1725    =/= wne 2598   RRcr 8981    +oocpnf 9109    -oocmnf 9110   RR*cxr 9111
This theorem is referenced by:  xaddnemnf  10812  xaddass  10820  xlesubadd  10834  xblss2ps  18423  xblss2  18424  nmoix  18755  nmoleub  18757  blcvx  18821  xrge0tsms  18857  metdstri  18873  nmoleub2lem  19114  xrge0nre  24205  xrge0tsmsd  24215
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9038  ax-resscn 9039
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-pnf 9114  df-mnf 9115  df-xr 9116
  Copyright terms: Public domain W3C validator