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Theorem lt0ne0 8183
Description: A number which is less than zero is not zero. See also lt0ap0 8403 which is similar but for apartness. (Contributed by Stefan O'Rear, 13-Sep-2014.)
Assertion
Ref Expression
lt0ne0  |-  ( ( A  e.  RR  /\  A  <  0 )  ->  A  =/=  0 )

Proof of Theorem lt0ne0
StepHypRef Expression
1 ltne 7842 . 2  |-  ( ( A  e.  RR  /\  A  <  0 )  -> 
0  =/=  A )
21necomd 2392 1  |-  ( ( A  e.  RR  /\  A  <  0 )  ->  A  =/=  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1480    =/= wne 2306   class class class wbr 3924   RRcr 7612   0cc0 7613    < clt 7793
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-pre-ltirr 7725
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-xp 4540  df-pnf 7795  df-mnf 7796  df-ltxr 7798
This theorem is referenced by: (None)
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