ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  subne0ad Unicode version

Theorem subne0ad 8292
Description: If the difference of two complex numbers is nonzero, they are unequal. Converse of subne0d 8290. Contrapositive of subeq0bd 8349. (Contributed by David Moews, 28-Feb-2017.)
Hypotheses
Ref Expression
negidd.1  |-  ( ph  ->  A  e.  CC )
pncand.2  |-  ( ph  ->  B  e.  CC )
subne0ad.3  |-  ( ph  ->  ( A  -  B
)  =/=  0 )
Assertion
Ref Expression
subne0ad  |-  ( ph  ->  A  =/=  B )

Proof of Theorem subne0ad
StepHypRef Expression
1 subne0ad.3 . 2  |-  ( ph  ->  ( A  -  B
)  =/=  0 )
2 negidd.1 . . . 4  |-  ( ph  ->  A  e.  CC )
3 pncand.2 . . . 4  |-  ( ph  ->  B  e.  CC )
42, 3subeq0ad 8291 . . 3  |-  ( ph  ->  ( ( A  -  B )  =  0  <-> 
A  =  B ) )
54necon3bid 2398 . 2  |-  ( ph  ->  ( ( A  -  B )  =/=  0  <->  A  =/=  B ) )
61, 5mpbid 147 1  |-  ( ph  ->  A  =/=  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 2158    =/= wne 2357  (class class class)co 5888   CCcc 7822   0cc0 7824    - cmin 8141
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-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-setind 4548  ax-resscn 7916  ax-1cn 7917  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-addcom 7924  ax-addass 7926  ax-distr 7928  ax-i2m1 7929  ax-0id 7932  ax-rnegex 7933  ax-cnre 7935
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-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  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-id 4305  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-iota 5190  df-fun 5230  df-fv 5236  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-sub 8143
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator