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Theorem xrmnfdc 9841
Description: An extended real is or is not minus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)
Assertion
Ref Expression
xrmnfdc  |-  ( A  e.  RR*  -> DECID  A  = -oo )

Proof of Theorem xrmnfdc
StepHypRef Expression
1 elxr 9774 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 renemnf 8004 . . . . . 6  |-  ( A  e.  RR  ->  A  =/= -oo )
32neneqd 2368 . . . . 5  |-  ( A  e.  RR  ->  -.  A  = -oo )
43olcd 734 . . . 4  |-  ( A  e.  RR  ->  ( A  = -oo  \/  -.  A  = -oo )
)
5 df-dc 835 . . . 4  |-  (DECID  A  = -oo  <->  ( A  = -oo  \/  -.  A  = -oo ) )
64, 5sylibr 134 . . 3  |-  ( A  e.  RR  -> DECID  A  = -oo )
7 pnfnemnf 8010 . . . . . . 7  |- +oo  =/= -oo
87neii 2349 . . . . . 6  |-  -. +oo  = -oo
9 eqeq1 2184 . . . . . 6  |-  ( A  = +oo  ->  ( A  = -oo  <-> +oo  = -oo ) )
108, 9mtbiri 675 . . . . 5  |-  ( A  = +oo  ->  -.  A  = -oo )
1110olcd 734 . . . 4  |-  ( A  = +oo  ->  ( A  = -oo  \/  -.  A  = -oo )
)
1211, 5sylibr 134 . . 3  |-  ( A  = +oo  -> DECID  A  = -oo )
13 orc 712 . . . 4  |-  ( A  = -oo  ->  ( A  = -oo  \/  -.  A  = -oo )
)
1413, 5sylibr 134 . . 3  |-  ( A  = -oo  -> DECID  A  = -oo )
156, 12, 143jaoi 1303 . 2  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  -> DECID  A  = -oo )
161, 15sylbi 121 1  |-  ( A  e.  RR*  -> DECID  A  = -oo )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 708  DECID wdc 834    \/ w3o 977    = wceq 1353    e. wcel 2148   RRcr 7809   +oocpnf 7987   -oocmnf 7988   RR*cxr 7989
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-uni 3810  df-pnf 7992  df-mnf 7993  df-xr 7994
This theorem is referenced by:  xaddf  9842  xaddval  9843  xaddmnf1  9846  xaddcom  9859  xnegdi  9866  xpncan  9869  xleadd1a  9871  xsubge0  9879  xrmaxiflemcl  11248  xrmaxifle  11249  xrmaxiflemab  11250  xrmaxiflemlub  11251  xrmaxiflemcom  11252  xrmaxadd  11264
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