Theorem lt0ne0d 8534

Description: Something less than zero is not zero. Deduction form. See also lt0ap0d 8670 which is similar but for apartness. (Contributed by David Moews, 28-Feb-2017.)

Hypothesis

Ref	Expression
lt0ne0d.1	⊢ (𝜑 → 𝐴 < 0)

Assertion

Ref	Expression
lt0ne0d	⊢ (𝜑 → 𝐴 ≠ 0)

Proof of Theorem lt0ne0d

Step	Hyp	Ref	Expression
1		lt0ne0d.1	. 2 ⊢ (𝜑 → 𝐴 < 0)
2		0re 8021	. . . . 5 ⊢ 0 ∈ ℝ
3	2	ltnri 8114	. . . 4 ⊢ ¬ 0 < 0
4		breq1 4033	. . . 4 ⊢ (𝐴 = 0 → (𝐴 < 0 ↔ 0 < 0))
5	3, 4	mtbiri 676	. . 3 ⊢ (𝐴 = 0 → ¬ 𝐴 < 0)
6	5	necon2ai 2418	. 2 ⊢ (𝐴 < 0 → 𝐴 ≠ 0)
7	1, 6	syl 14	1 ⊢ (𝜑 → 𝐴 ≠ 0)

Syntax hints:   → wi 4   = wceq 1364   ≠ wne 2364   class class class wbr 4030  0cc0 7874   < clt 8056

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 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971  ax-rnegex 7983  ax-pre-ltirr 7986

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 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-xp 4666  df-pnf 8058  df-mnf 8059  df-ltxr 8061

This theorem is referenced by:  divalglemeuneg  12067