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Theorem lt0ne0 8212
Description: A number which is less than zero is not zero. See also lt0ap0 8432 which is similar but for apartness. (Contributed by Stefan O'Rear, 13-Sep-2014.)
Assertion
Ref Expression
lt0ne0 ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 ≠ 0)

Proof of Theorem lt0ne0
StepHypRef Expression
1 ltne 7871 . 2 ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 0 ≠ 𝐴)
21necomd 2395 1 ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 ≠ 0)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wcel 1481  wne 2309   class class class wbr 3935  cr 7641  0cc0 7642   < clt 7822
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 4052  ax-pow 4104  ax-pr 4137  ax-un 4361  ax-setind 4458  ax-cnex 7733  ax-resscn 7734  ax-pre-ltirr 7754
This theorem depends on definitions:  df-bi 116  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 2691  df-dif 3076  df-un 3078  df-in 3080  df-ss 3087  df-pw 3515  df-sn 3536  df-pr 3537  df-op 3539  df-uni 3743  df-br 3936  df-opab 3996  df-xp 4551  df-pnf 7824  df-mnf 7825  df-ltxr 7827
This theorem is referenced by: (None)
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