Theorem lt0ne0d 8275

Description: Something less than zero is not zero. Deduction form. See also lt0ap0d 8411 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 7766	. . . . 5 |- 0 e. RR
3	2	ltnri 7856	. . . 4 |- -. 0 < 0
4		breq1 3932	. . . 4 |- ( A = 0 -> ( A < 0 <-> 0 < 0 ) )
5	3, 4	mtbiri 664	. . 3 |- ( A = 0 -> -. A < 0 )
6	5	necon2ai 2362	. 2 |- ( A < 0 -> A =/= 0 )
7	1, 6	syl 14	1 |- ( ph -> A =/= 0 )

Syntax hints:   -> wi 4   = wceq 1331   =/= wne 2308   class class class wbr 3929   0cc0 7620   < clt 7800

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1re 7714  ax-addrcl 7717  ax-rnegex 7729  ax-pre-ltirr 7732

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-pnf 7802  df-mnf 7803  df-ltxr 7805

This theorem is referenced by:  divalglemeuneg  11620