Theorem lt0ne0d 8411

Description: Something less than zero is not zero. Deduction form. See also lt0ap0d 8547 which is similar but for apartness. (Contributed by David Moews, 28-Feb-2017.)

Hypothesis

Ref	Expression
lt0ne0d.1	|- ( ph -> A < 0 )

Assertion

Ref	Expression
lt0ne0d	|- ( ph -> A =/= 0 )

Proof of Theorem lt0ne0d

Step	Hyp	Ref	Expression
1		lt0ne0d.1	. 2 |- ( ph -> A < 0 )
2		0re 7899	. . . . 5 |- 0 e. RR
3	2	ltnri 7991	. . . 4 |- -. 0 < 0
4		breq1 3985	. . . 4 |- ( A = 0 -> ( A < 0 <-> 0 < 0 ) )
5	3, 4	mtbiri 665	. . 3 |- ( A = 0 -> -. A < 0 )
6	5	necon2ai 2390	. 2 |- ( A < 0 -> A =/= 0 )
7	1, 6	syl 14	1 |- ( ph -> A =/= 0 )

Syntax hints:   -> wi 4   = wceq 1343   =/= wne 2336   class class class wbr 3982   0cc0 7753   < clt 7933

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1re 7847  ax-addrcl 7850  ax-rnegex 7862  ax-pre-ltirr 7865

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-pnf 7935  df-mnf 7936  df-ltxr 7938

This theorem is referenced by:  divalglemeuneg  11860