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Theorem npnflt 9624
Description: An extended real is less than plus infinity iff they are not equal. (Contributed by Jim Kingdon, 17-Apr-2023.)
Assertion
Ref Expression
npnflt  |-  ( A  e.  RR*  ->  ( A  < +oo  <->  A  =/= +oo )
)

Proof of Theorem npnflt
StepHypRef Expression
1 nltpnft 9623 . . . 4  |-  ( A  e.  RR*  ->  ( A  = +oo  <->  -.  A  < +oo ) )
21biimpd 143 . . 3  |-  ( A  e.  RR*  ->  ( A  = +oo  ->  -.  A  < +oo ) )
32necon2ad 2366 . 2  |-  ( A  e.  RR*  ->  ( A  < +oo  ->  A  =/= +oo ) )
4 ltpnf 9593 . . . . 5  |-  ( A  e.  RR  ->  A  < +oo )
54adantl 275 . . . 4  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  A  e.  RR )  ->  A  < +oo )
6 simpr 109 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  A  = +oo )  ->  A  = +oo )
7 simplr 520 . . . . 5  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  A  = +oo )  ->  A  =/= +oo )
86, 7pm2.21ddne 2392 . . . 4  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  A  = +oo )  ->  A  < +oo )
9 mnfltpnf 9597 . . . . . 6  |- -oo  < +oo
10 breq1 3936 . . . . . 6  |-  ( A  = -oo  ->  ( A  < +oo  <-> -oo  < +oo )
)
119, 10mpbiri 167 . . . . 5  |-  ( A  = -oo  ->  A  < +oo )
1211adantl 275 . . . 4  |-  ( ( ( A  e.  RR*  /\  A  =/= +oo )  /\  A  = -oo )  ->  A  < +oo )
13 elxr 9589 . . . . . 6  |-  ( A  e.  RR*  <->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
1413biimpi 119 . . . . 5  |-  ( A  e.  RR*  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
1514adantr 274 . . . 4  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  ( A  e.  RR  \/  A  = +oo  \/  A  = -oo ) )
165, 8, 12, 15mpjao3dan 1286 . . 3  |-  ( ( A  e.  RR*  /\  A  =/= +oo )  ->  A  < +oo )
1716ex 114 . 2  |-  ( A  e.  RR*  ->  ( A  =/= +oo  ->  A  < +oo ) )
183, 17impbid 128 1  |-  ( A  e.  RR*  ->  ( A  < +oo  <->  A  =/= +oo )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ w3o 962    = wceq 1332    e. wcel 1481    =/= wne 2309   class class class wbr 3933   RRcr 7639   +oocpnf 7817   -oocmnf 7818   RR*cxr 7819    < clt 7820
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-setind 4456  ax-cnex 7731  ax-resscn 7732  ax-pre-ltirr 7752
This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2689  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-br 3934  df-opab 3994  df-xp 4549  df-pnf 7822  df-mnf 7823  df-xr 7824  df-ltxr 7825
This theorem is referenced by:  xlt2add  9689  xrmaxadd  11058
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