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Theorem lt0ne0d 8505
Description: Something less than zero is not zero. Deduction form. See also lt0ap0d 8641 which is similar but for apartness. (Contributed by David Moews, 28-Feb-2017.)
Hypothesis
Ref Expression
lt0ne0d.1 (𝜑𝐴 < 0)
Assertion
Ref Expression
lt0ne0d (𝜑𝐴 ≠ 0)

Proof of Theorem lt0ne0d
StepHypRef Expression
1 lt0ne0d.1 . 2 (𝜑𝐴 < 0)
2 0re 7992 . . . . 5 0 ∈ ℝ
32ltnri 8085 . . . 4 ¬ 0 < 0
4 breq1 4024 . . . 4 (𝐴 = 0 → (𝐴 < 0 ↔ 0 < 0))
53, 4mtbiri 676 . . 3 (𝐴 = 0 → ¬ 𝐴 < 0)
65necon2ai 2414 . 2 (𝐴 < 0 → 𝐴 ≠ 0)
71, 6syl 14 1 (𝜑𝐴 ≠ 0)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  wne 2360   class class class wbr 4021  0cc0 7846   < clt 8027
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 4139  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-cnex 7937  ax-resscn 7938  ax-1re 7940  ax-addrcl 7943  ax-rnegex 7955  ax-pre-ltirr 7958
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 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-xp 4653  df-pnf 8029  df-mnf 8030  df-ltxr 8032
This theorem is referenced by:  divalglemeuneg  11969
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