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Theorem nelne1 2336
Description: Two classes are different if they don't contain the same element. (Contributed by NM, 3-Feb-2012.)
Assertion
Ref Expression
nelne1  |-  ( ( A  e.  B  /\  -.  A  e.  C
)  ->  B  =/=  C )

Proof of Theorem nelne1
StepHypRef Expression
1 eleq2 2143 . . . 4  |-  ( B  =  C  ->  ( A  e.  B  <->  A  e.  C ) )
21biimpcd 157 . . 3  |-  ( A  e.  B  ->  ( B  =  C  ->  A  e.  C ) )
32necon3bd 2289 . 2  |-  ( A  e.  B  ->  ( -.  A  e.  C  ->  B  =/=  C ) )
43imp 122 1  |-  ( ( A  e.  B  /\  -.  A  e.  C
)  ->  B  =/=  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434    =/= wne 2246
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-17 1460  ax-ial 1468  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-cleq 2075  df-clel 2078  df-ne 2247
This theorem is referenced by:  difsnb  3536
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