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Theorem ngtmnft 9593
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 9556 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 renemnf 7807 . . . . 5  |-  ( A  e.  RR  ->  A  =/= -oo )
32neneqd 2327 . . . 4  |-  ( A  e.  RR  ->  -.  A  = -oo )
4 mnflt 9562 . . . . 5  |-  ( A  e.  RR  -> -oo  <  A )
5 notnot 618 . . . . 5  |-  ( -oo  <  A  ->  -.  -. -oo  <  A )
64, 5syl 14 . . . 4  |-  ( A  e.  RR  ->  -.  -. -oo  <  A )
73, 62falsed 691 . . 3  |-  ( A  e.  RR  ->  ( A  = -oo  <->  -. -oo  <  A ) )
8 pnfnemnf 7813 . . . . . 6  |- +oo  =/= -oo
9 neeq1 2319 . . . . . 6  |-  ( A  = +oo  ->  ( A  =/= -oo  <-> +oo  =/= -oo )
)
108, 9mpbiri 167 . . . . 5  |-  ( A  = +oo  ->  A  =/= -oo )
1110neneqd 2327 . . . 4  |-  ( A  = +oo  ->  -.  A  = -oo )
12 mnfltpnf 9564 . . . . . . 7  |- -oo  < +oo
13 breq2 3928 . . . . . . 7  |-  ( A  = +oo  ->  ( -oo  <  A  <-> -oo  < +oo ) )
1412, 13mpbiri 167 . . . . . 6  |-  ( A  = +oo  -> -oo  <  A )
1514necon3bi 2356 . . . . 5  |-  ( -. -oo  <  A  ->  A  =/= +oo )
1615necon2bi 2361 . . . 4  |-  ( A  = +oo  ->  -.  -. -oo  <  A )
1711, 162falsed 691 . . 3  |-  ( A  = +oo  ->  ( A  = -oo  <->  -. -oo  <  A ) )
18 id 19 . . . 4  |-  ( A  = -oo  ->  A  = -oo )
19 mnfxr 7815 . . . . . 6  |- -oo  e.  RR*
20 xrltnr 9559 . . . . . 6  |-  ( -oo  e.  RR*  ->  -. -oo  < -oo )
2119, 20ax-mp 5 . . . . 5  |-  -. -oo  < -oo
22 breq2 3928 . . . . 5  |-  ( A  = -oo  ->  ( -oo  <  A  <-> -oo  < -oo ) )
2321, 22mtbiri 664 . . . 4  |-  ( A  = -oo  ->  -. -oo 
<  A )
2418, 232thd 174 . . 3  |-  ( A  = -oo  ->  ( A  = -oo  <->  -. -oo  <  A ) )
257, 17, 243jaoi 1281 . 2  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  ( A  = -oo  <->  -. -oo  <  A ) )
261, 25sylbi 120 1  |-  ( A  e.  RR*  ->  ( A  = -oo  <->  -. -oo  <  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104    \/ w3o 961    = wceq 1331    e. wcel 1480    =/= wne 2306   class class class wbr 3924   RRcr 7612   +oocpnf 7790   -oocmnf 7791   RR*cxr 7792    < clt 7793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-pre-ltirr 7725
This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-xp 4540  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798
This theorem is referenced by:  nmnfgt  9594  ge0nemnf  9600  xleaddadd  9663
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