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

Theorem neqned 2256
Description: If it is not the case that two classes are equal, they are unequal. Converse of neneqd 2270. One-way deduction form of df-ne 2250. (Contributed by David Moews, 28-Feb-2017.) Allow a shortening of necon3bi 2299. (Revised by Wolf Lammen, 22-Nov-2019.)
Hypothesis
Ref Expression
neqned.1  |-  ( ph  ->  -.  A  =  B )
Assertion
Ref Expression
neqned  |-  ( ph  ->  A  =/=  B )

Proof of Theorem neqned
StepHypRef Expression
1 neqned.1 . 2  |-  ( ph  ->  -.  A  =  B )
2 df-ne 2250 . 2  |-  ( A  =/=  B  <->  -.  A  =  B )
31, 2sylibr 132 1  |-  ( ph  ->  A  =/=  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1285    =/= wne 2249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106
This theorem depends on definitions:  df-bi 115  df-ne 2250
This theorem is referenced by:  neqne  2257  tfr1onlemsucaccv  6038  tfrcllemsucaccv  6051  djune  6676  flqltnz  9583  bezoutlemle  10777  eucalgval2  10815  eucalglt  10819  lcmval  10825  lcmcllem  10829  isprm2  10879  sqne2sq  10935
  Copyright terms: Public domain W3C validator