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Theorem nltpnft 9771
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 9733 . 2  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
2 renepnf 7967 . . . . 5  |-  ( A  e.  RR  ->  A  =/= +oo )
32neneqd 2361 . . . 4  |-  ( A  e.  RR  ->  -.  A  = +oo )
4 ltpnf 9737 . . . . 5  |-  ( A  e.  RR  ->  A  < +oo )
5 notnot 624 . . . . 5  |-  ( A  < +oo  ->  -.  -.  A  < +oo )
64, 5syl 14 . . . 4  |-  ( A  e.  RR  ->  -.  -.  A  < +oo )
73, 62falsed 697 . . 3  |-  ( A  e.  RR  ->  ( A  = +oo  <->  -.  A  < +oo ) )
8 id 19 . . . 4  |-  ( A  = +oo  ->  A  = +oo )
9 pnfxr 7972 . . . . . 6  |- +oo  e.  RR*
10 xrltnr 9736 . . . . . 6  |-  ( +oo  e.  RR*  ->  -. +oo  < +oo )
119, 10ax-mp 5 . . . . 5  |-  -. +oo  < +oo
12 breq1 3992 . . . . 5  |-  ( A  = +oo  ->  ( A  < +oo  <-> +oo  < +oo )
)
1311, 12mtbiri 670 . . . 4  |-  ( A  = +oo  ->  -.  A  < +oo )
148, 132thd 174 . . 3  |-  ( A  = +oo  ->  ( A  = +oo  <->  -.  A  < +oo ) )
15 mnfnepnf 7975 . . . . . 6  |- -oo  =/= +oo
1615neii 2342 . . . . 5  |-  -. -oo  = +oo
17 eqeq1 2177 . . . . 5  |-  ( A  = -oo  ->  ( A  = +oo  <-> -oo  = +oo ) )
1816, 17mtbiri 670 . . . 4  |-  ( A  = -oo  ->  -.  A  = +oo )
19 mnfltpnf 9742 . . . . . . 7  |- -oo  < +oo
20 breq1 3992 . . . . . . 7  |-  ( A  = -oo  ->  ( A  < +oo  <-> -oo  < +oo )
)
2119, 20mpbiri 167 . . . . . 6  |-  ( A  = -oo  ->  A  < +oo )
2221necon3bi 2390 . . . . 5  |-  ( -.  A  < +oo  ->  A  =/= -oo )
2322necon2bi 2395 . . . 4  |-  ( A  = -oo  ->  -.  -.  A  < +oo )
2418, 232falsed 697 . . 3  |-  ( A  = -oo  ->  ( A  = +oo  <->  -.  A  < +oo ) )
257, 14, 243jaoi 1298 . 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 972    = wceq 1348    e. wcel 2141   class class class wbr 3989   RRcr 7773   +oocpnf 7951   -oocmnf 7952   RR*cxr 7953    < clt 7954
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-pre-ltirr 7886
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959
This theorem is referenced by:  npnflt  9772  xgepnf  9773  xrmaxiflemlub  11211
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