Theorem lt0ne0 8571

Description: A number which is less than zero is not zero. See also lt0ap0 8791 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

Step	Hyp	Ref	Expression
1		ltne 8227	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 0 ≠ 𝐴)
2	1	necomd 2486	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 < 0) → 𝐴 ≠ 0)

Syntax hints:   → wi 4   ∧ wa 104   ∈ wcel 2200   ≠ wne 2400   class class class wbr 4082  ℝcr 7994  0cc0 7995   < clt 8177

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-cnex 8086  ax-resscn 8087  ax-pre-ltirr 8107

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-xp 4724  df-pnf 8179  df-mnf 8180  df-ltxr 8182

This theorem is referenced by: (None)