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Theorem lt0ne0 8414
Description: A number which is less than zero is not zero. See also lt0ap0 8634 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 8071 . 2 ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 0 ≠ 𝐴)
21necomd 2446 1 ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 ≠ 0)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wcel 2160  wne 2360   class class class wbr 4018  cr 7839  0cc0 7840   < clt 8021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-cnex 7931  ax-resscn 7932  ax-pre-ltirr 7952
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-xp 4650  df-pnf 8023  df-mnf 8024  df-ltxr 8026
This theorem is referenced by: (None)
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