Theorem lt0ne0d 8432

Description: Something less than zero is not zero. Deduction form. See also lt0ap0d 8568 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 7920	. . . . 5 |- 0 e. RR
3	2	ltnri 8012	. . . 4 |- -. 0 < 0
4		breq1 3992	. . . 4 |- ( A = 0 -> ( A < 0 <-> 0 < 0 ) )
5	3, 4	mtbiri 670	. . 3 |- ( A = 0 -> -. A < 0 )
6	5	necon2ai 2394	. 2 |- ( A < 0 -> A =/= 0 )
7	1, 6	syl 14	1 |- ( ph -> A =/= 0 )

Syntax hints:   -> wi 4   = wceq 1348   =/= wne 2340   class class class wbr 3989   0cc0 7774   < clt 7954

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-1re 7868  ax-addrcl 7871  ax-rnegex 7883  ax-pre-ltirr 7886

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-xp 4617  df-pnf 7956  df-mnf 7957  df-ltxr 7959

This theorem is referenced by:  divalglemeuneg  11882