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Theorem ngtmnft 9278
Description: An extended real is not greater than minus infinity iff they are equal. (Contributed by NM, 2-Feb-2006.)
Assertion
Ref Expression
ngtmnft  |-  ( A  e.  RR*  ->  ( A  = -oo  <->  -. -oo  <  A ) )

Proof of Theorem ngtmnft
StepHypRef Expression
1 elxr 9245 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 renemnf 7534 . . . . 5  |-  ( A  e.  RR  ->  A  =/= -oo )
32neneqd 2276 . . . 4  |-  ( A  e.  RR  ->  -.  A  = -oo )
4 mnflt 9251 . . . . 5  |-  ( A  e.  RR  -> -oo  <  A )
5 notnot 594 . . . . 5  |-  ( -oo  <  A  ->  -.  -. -oo  <  A )
64, 5syl 14 . . . 4  |-  ( A  e.  RR  ->  -.  -. -oo  <  A )
73, 62falsed 653 . . 3  |-  ( A  e.  RR  ->  ( A  = -oo  <->  -. -oo  <  A ) )
8 pnfnemnf 7540 . . . . . 6  |- +oo  =/= -oo
9 neeq1 2268 . . . . . 6  |-  ( A  = +oo  ->  ( A  =/= -oo  <-> +oo  =/= -oo )
)
108, 9mpbiri 166 . . . . 5  |-  ( A  = +oo  ->  A  =/= -oo )
1110neneqd 2276 . . . 4  |-  ( A  = +oo  ->  -.  A  = -oo )
12 mnfltpnf 9253 . . . . . . 7  |- -oo  < +oo
13 breq2 3849 . . . . . . 7  |-  ( A  = +oo  ->  ( -oo  <  A  <-> -oo  < +oo ) )
1412, 13mpbiri 166 . . . . . 6  |-  ( A  = +oo  -> -oo  <  A )
1514necon3bi 2305 . . . . 5  |-  ( -. -oo  <  A  ->  A  =/= +oo )
1615necon2bi 2310 . . . 4  |-  ( A  = +oo  ->  -.  -. -oo  <  A )
1711, 162falsed 653 . . 3  |-  ( A  = +oo  ->  ( A  = -oo  <->  -. -oo  <  A ) )
18 id 19 . . . 4  |-  ( A  = -oo  ->  A  = -oo )
19 mnfxr 7542 . . . . . 6  |- -oo  e.  RR*
20 xrltnr 9248 . . . . . 6  |-  ( -oo  e.  RR*  ->  -. -oo  < -oo )
2119, 20ax-mp 7 . . . . 5  |-  -. -oo  < -oo
22 breq2 3849 . . . . 5  |-  ( A  = -oo  ->  ( -oo  <  A  <-> -oo  < -oo ) )
2321, 22mtbiri 635 . . . 4  |-  ( A  = -oo  ->  -. -oo 
<  A )
2418, 232thd 173 . . 3  |-  ( A  = -oo  ->  ( A  = -oo  <->  -. -oo  <  A ) )
257, 17, 243jaoi 1239 . 2  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  ( A  = -oo  <->  -. -oo  <  A ) )
261, 25sylbi 119 1  |-  ( A  e.  RR*  ->  ( A  = -oo  <->  -. -oo  <  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 103    \/ w3o 923    = wceq 1289    e. wcel 1438    =/= wne 2255   class class class wbr 3845   RRcr 7347   +oocpnf 7517   -oocmnf 7518   RR*cxr 7519    < clt 7520
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260  ax-setind 4353  ax-cnex 7434  ax-resscn 7435  ax-pre-ltirr 7455
This theorem depends on definitions:  df-bi 115  df-3or 925  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 3001  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-xp 4444  df-pnf 7522  df-mnf 7523  df-xr 7524  df-ltxr 7525
This theorem is referenced by:  ge0nemnf  9284
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