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Theorem ngtmnft 9721
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 9683 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 renemnf 7926 . . . . 5  |-  ( A  e.  RR  ->  A  =/= -oo )
32neneqd 2348 . . . 4  |-  ( A  e.  RR  ->  -.  A  = -oo )
4 mnflt 9690 . . . . 5  |-  ( A  e.  RR  -> -oo  <  A )
5 notnot 619 . . . . 5  |-  ( -oo  <  A  ->  -.  -. -oo  <  A )
64, 5syl 14 . . . 4  |-  ( A  e.  RR  ->  -.  -. -oo  <  A )
73, 62falsed 692 . . 3  |-  ( A  e.  RR  ->  ( A  = -oo  <->  -. -oo  <  A ) )
8 pnfnemnf 7932 . . . . . 6  |- +oo  =/= -oo
9 neeq1 2340 . . . . . 6  |-  ( A  = +oo  ->  ( A  =/= -oo  <-> +oo  =/= -oo )
)
108, 9mpbiri 167 . . . . 5  |-  ( A  = +oo  ->  A  =/= -oo )
1110neneqd 2348 . . . 4  |-  ( A  = +oo  ->  -.  A  = -oo )
12 mnfltpnf 9692 . . . . . . 7  |- -oo  < +oo
13 breq2 3969 . . . . . . 7  |-  ( A  = +oo  ->  ( -oo  <  A  <-> -oo  < +oo ) )
1412, 13mpbiri 167 . . . . . 6  |-  ( A  = +oo  -> -oo  <  A )
1514necon3bi 2377 . . . . 5  |-  ( -. -oo  <  A  ->  A  =/= +oo )
1615necon2bi 2382 . . . 4  |-  ( A  = +oo  ->  -.  -. -oo  <  A )
1711, 162falsed 692 . . 3  |-  ( A  = +oo  ->  ( A  = -oo  <->  -. -oo  <  A ) )
18 id 19 . . . 4  |-  ( A  = -oo  ->  A  = -oo )
19 mnfxr 7934 . . . . . 6  |- -oo  e.  RR*
20 xrltnr 9686 . . . . . 6  |-  ( -oo  e.  RR*  ->  -. -oo  < -oo )
2119, 20ax-mp 5 . . . . 5  |-  -. -oo  < -oo
22 breq2 3969 . . . . 5  |-  ( A  = -oo  ->  ( -oo  <  A  <-> -oo  < -oo ) )
2321, 22mtbiri 665 . . . 4  |-  ( A  = -oo  ->  -. -oo 
<  A )
2418, 232thd 174 . . 3  |-  ( A  = -oo  ->  ( A  = -oo  <->  -. -oo  <  A ) )
257, 17, 243jaoi 1285 . 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 962    = wceq 1335    e. wcel 2128    =/= wne 2327   class class class wbr 3965   RRcr 7731   +oocpnf 7909   -oocmnf 7910   RR*cxr 7911    < clt 7912
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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-cnex 7823  ax-resscn 7824  ax-pre-ltirr 7844
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-nel 2423  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-xp 4592  df-pnf 7914  df-mnf 7915  df-xr 7916  df-ltxr 7917
This theorem is referenced by:  nmnfgt  9722  ge0nemnf  9728  xleaddadd  9791
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