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Theorem lt0ne0d 8243
Description: Something less than zero is not zero. Deduction form. See also lt0ap0d 8379 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 7734 . . . . 5 0 ∈ ℝ
32ltnri 7824 . . . 4 ¬ 0 < 0
4 breq1 3902 . . . 4 (𝐴 = 0 → (𝐴 < 0 ↔ 0 < 0))
53, 4mtbiri 649 . . 3 (𝐴 = 0 → ¬ 𝐴 < 0)
65necon2ai 2339 . 2 (𝐴 < 0 → 𝐴 ≠ 0)
71, 6syl 14 1 (𝜑𝐴 ≠ 0)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1316  wne 2285   class class class wbr 3899  0cc0 7588   < clt 7768
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 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1re 7682  ax-addrcl 7685  ax-rnegex 7697  ax-pre-ltirr 7700
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-rab 2402  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-xp 4515  df-pnf 7770  df-mnf 7771  df-ltxr 7773
This theorem is referenced by:  divalglemeuneg  11547
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