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Theorem nltpnft 9744
Description: An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.)
Assertion
Ref Expression
nltpnft  |-  ( A  e.  RR*  ->  ( A  = +oo  <->  -.  A  < +oo ) )

Proof of Theorem nltpnft
StepHypRef Expression
1 elxr 9706 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 renepnf 7940 . . . . 5  |-  ( A  e.  RR  ->  A  =/= +oo )
32neneqd 2355 . . . 4  |-  ( A  e.  RR  ->  -.  A  = +oo )
4 ltpnf 9710 . . . . 5  |-  ( A  e.  RR  ->  A  < +oo )
5 notnot 619 . . . . 5  |-  ( A  < +oo  ->  -.  -.  A  < +oo )
64, 5syl 14 . . . 4  |-  ( A  e.  RR  ->  -.  -.  A  < +oo )
73, 62falsed 692 . . 3  |-  ( A  e.  RR  ->  ( A  = +oo  <->  -.  A  < +oo ) )
8 id 19 . . . 4  |-  ( A  = +oo  ->  A  = +oo )
9 pnfxr 7945 . . . . . 6  |- +oo  e.  RR*
10 xrltnr 9709 . . . . . 6  |-  ( +oo  e.  RR*  ->  -. +oo  < +oo )
119, 10ax-mp 5 . . . . 5  |-  -. +oo  < +oo
12 breq1 3982 . . . . 5  |-  ( A  = +oo  ->  ( A  < +oo  <-> +oo  < +oo )
)
1311, 12mtbiri 665 . . . 4  |-  ( A  = +oo  ->  -.  A  < +oo )
148, 132thd 174 . . 3  |-  ( A  = +oo  ->  ( A  = +oo  <->  -.  A  < +oo ) )
15 mnfnepnf 7948 . . . . . 6  |- -oo  =/= +oo
1615neii 2336 . . . . 5  |-  -. -oo  = +oo
17 eqeq1 2171 . . . . 5  |-  ( A  = -oo  ->  ( A  = +oo  <-> -oo  = +oo ) )
1816, 17mtbiri 665 . . . 4  |-  ( A  = -oo  ->  -.  A  = +oo )
19 mnfltpnf 9715 . . . . . . 7  |- -oo  < +oo
20 breq1 3982 . . . . . . 7  |-  ( A  = -oo  ->  ( A  < +oo  <-> -oo  < +oo )
)
2119, 20mpbiri 167 . . . . . 6  |-  ( A  = -oo  ->  A  < +oo )
2221necon3bi 2384 . . . . 5  |-  ( -.  A  < +oo  ->  A  =/= -oo )
2322necon2bi 2389 . . . 4  |-  ( A  = -oo  ->  -.  -.  A  < +oo )
2418, 232falsed 692 . . 3  |-  ( A  = -oo  ->  ( A  = +oo  <->  -.  A  < +oo ) )
257, 14, 243jaoi 1292 . 2  |-  ( ( A  e.  RR  \/  A  = +oo  \/  A  = -oo )  ->  ( A  = +oo  <->  -.  A  < +oo ) )
261, 25sylbi 120 1  |-  ( A  e.  RR*  ->  ( A  = +oo  <->  -.  A  < +oo ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 104    \/ w3o 966    = wceq 1342    e. wcel 2135   class class class wbr 3979   RRcr 7746   +oocpnf 7924   -oocmnf 7925   RR*cxr 7926    < clt 7927
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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4097  ax-pow 4150  ax-pr 4184  ax-un 4408  ax-setind 4511  ax-cnex 7838  ax-resscn 7839  ax-pre-ltirr 7859
This theorem depends on definitions:  df-bi 116  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2726  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pw 3558  df-sn 3579  df-pr 3580  df-op 3582  df-uni 3787  df-br 3980  df-opab 4041  df-xp 4607  df-pnf 7929  df-mnf 7930  df-xr 7931  df-ltxr 7932
This theorem is referenced by:  npnflt  9745  xgepnf  9746  xrmaxiflemlub  11183
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