ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  nelne2 Unicode version

Theorem nelne2 2376
Description: Two classes are different if they don't belong to the same class. (Contributed by NM, 25-Jun-2012.)
Assertion
Ref Expression
nelne2  |-  ( ( A  e.  C  /\  -.  B  e.  C
)  ->  A  =/=  B )

Proof of Theorem nelne2
StepHypRef Expression
1 eleq1 2180 . . . 4  |-  ( A  =  B  ->  ( A  e.  C  <->  B  e.  C ) )
21biimpcd 158 . . 3  |-  ( A  e.  C  ->  ( A  =  B  ->  B  e.  C ) )
32necon3bd 2328 . 2  |-  ( A  e.  C  ->  ( -.  B  e.  C  ->  A  =/=  B ) )
43imp 123 1  |-  ( ( A  e.  C  /\  -.  B  e.  C
)  ->  A  =/=  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465    =/= wne 2285
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 588  ax-in2 589  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-ial 1499  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-cleq 2110  df-clel 2113  df-ne 2286
This theorem is referenced by:  nelelne  2377  elnelne2  2390
  Copyright terms: Public domain W3C validator