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Theorem ngtmnft 9939
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 9898 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 renemnf 8121 . . . . 5  |-  ( A  e.  RR  ->  A  =/= -oo )
32neneqd 2397 . . . 4  |-  ( A  e.  RR  ->  -.  A  = -oo )
4 mnflt 9905 . . . . 5  |-  ( A  e.  RR  -> -oo  <  A )
5 notnot 630 . . . . 5  |-  ( -oo  <  A  ->  -.  -. -oo  <  A )
64, 5syl 14 . . . 4  |-  ( A  e.  RR  ->  -.  -. -oo  <  A )
73, 62falsed 704 . . 3  |-  ( A  e.  RR  ->  ( A  = -oo  <->  -. -oo  <  A ) )
8 pnfnemnf 8127 . . . . . 6  |- +oo  =/= -oo
9 neeq1 2389 . . . . . 6  |-  ( A  = +oo  ->  ( A  =/= -oo  <-> +oo  =/= -oo )
)
108, 9mpbiri 168 . . . . 5  |-  ( A  = +oo  ->  A  =/= -oo )
1110neneqd 2397 . . . 4  |-  ( A  = +oo  ->  -.  A  = -oo )
12 mnfltpnf 9907 . . . . . . 7  |- -oo  < +oo
13 breq2 4048 . . . . . . 7  |-  ( A  = +oo  ->  ( -oo  <  A  <-> -oo  < +oo ) )
1412, 13mpbiri 168 . . . . . 6  |-  ( A  = +oo  -> -oo  <  A )
1514necon3bi 2426 . . . . 5  |-  ( -. -oo  <  A  ->  A  =/= +oo )
1615necon2bi 2431 . . . 4  |-  ( A  = +oo  ->  -.  -. -oo  <  A )
1711, 162falsed 704 . . 3  |-  ( A  = +oo  ->  ( A  = -oo  <->  -. -oo  <  A ) )
18 id 19 . . . 4  |-  ( A  = -oo  ->  A  = -oo )
19 mnfxr 8129 . . . . . 6  |- -oo  e.  RR*
20 xrltnr 9901 . . . . . 6  |-  ( -oo  e.  RR*  ->  -. -oo  < -oo )
2119, 20ax-mp 5 . . . . 5  |-  -. -oo  < -oo
22 breq2 4048 . . . . 5  |-  ( A  = -oo  ->  ( -oo  <  A  <-> -oo  < -oo ) )
2321, 22mtbiri 677 . . . 4  |-  ( A  = -oo  ->  -. -oo 
<  A )
2418, 232thd 175 . . 3  |-  ( A  = -oo  ->  ( A  = -oo  <->  -. -oo  <  A ) )
257, 17, 243jaoi 1316 . 2  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  ( A  = -oo  <->  -. -oo  <  A ) )
261, 25sylbi 121 1  |-  ( A  e.  RR*  ->  ( A  = -oo  <->  -. -oo  <  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 105    \/ w3o 980    = wceq 1373    e. wcel 2176    =/= wne 2376   class class class wbr 4044   RRcr 7924   +oocpnf 8104   -oocmnf 8105   RR*cxr 8106    < clt 8107
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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-pre-ltirr 8037
This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-opab 4106  df-xp 4681  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112
This theorem is referenced by:  nmnfgt  9940  ge0nemnf  9946  xleaddadd  10009
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