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Theorem lt0ne0d 7505
Description: Something less than zero is not zero. Deduction form. (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
StepHypRef Expression
1 lt0ne0d.1 . 2  |-  ( ph  ->  A  <  0 )
2 0re 7027 . . . . 5  |-  0  e.  RR
32ltnri 7110 . . . 4  |-  -.  0  <  0
4 breq1 3767 . . . 4  |-  ( A  =  0  ->  ( A  <  0  <->  0  <  0 ) )
53, 4mtbiri 600 . . 3  |-  ( A  =  0  ->  -.  A  <  0 )
65necon2ai 2259 . 2  |-  ( A  <  0  ->  A  =/=  0 )
71, 6syl 14 1  |-  ( ph  ->  A  =/=  0 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1243    =/= wne 2204   class class class wbr 3764   0cc0 6889    < clt 7060
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-1re 6978  ax-addrcl 6981  ax-rnegex 6993  ax-pre-ltirr 6996
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-pnf 7062  df-mnf 7063  df-ltxr 7065
This theorem is referenced by: (None)
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