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Theorem nelelne 2377
Description: Two classes are different if they don't belong to the same class. (Contributed by Rodolfo Medina, 17-Oct-2010.) (Proof shortened by AV, 10-May-2020.)
Assertion
Ref Expression
nelelne  |-  ( -.  A  e.  B  -> 
( C  e.  B  ->  C  =/=  A ) )

Proof of Theorem nelelne
StepHypRef Expression
1 nelne2 2376 . 2  |-  ( ( C  e.  B  /\  -.  A  e.  B
)  ->  C  =/=  A )
21expcom 115 1  |-  ( -.  A  e.  B  -> 
( C  e.  B  ->  C  =/=  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    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: (None)
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