Theorem lt0ne0d 8484

Description: Something less than zero is not zero. Deduction form. See also lt0ap0d 8620 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 7971	. . . . 5 |- 0 e. RR
3	2	ltnri 8064	. . . 4 |- -. 0 < 0
4		breq1 4018	. . . 4 |- ( A = 0 -> ( A < 0 <-> 0 < 0 ) )
5	3, 4	mtbiri 676	. . 3 |- ( A = 0 -> -. A < 0 )
6	5	necon2ai 2411	. 2 |- ( A < 0 -> A =/= 0 )
7	1, 6	syl 14	1 |- ( ph -> A =/= 0 )

Syntax hints:   -> wi 4   = wceq 1363   =/= wne 2357   class class class wbr 4015   0cc0 7825   < clt 8006

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7916  ax-resscn 7917  ax-1re 7919  ax-addrcl 7922  ax-rnegex 7934  ax-pre-ltirr 7937

This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-xp 4644  df-pnf 8008  df-mnf 8009  df-ltxr 8011

This theorem is referenced by:  divalglemeuneg  11942