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Theorem elnelne2 2411
Description: Two classes are different if they don't belong to the same class. (Contributed by AV, 28-Jan-2020.)
Assertion
Ref Expression
elnelne2  |-  ( ( A  e.  C  /\  B  e/  C )  ->  A  =/=  B )

Proof of Theorem elnelne2
StepHypRef Expression
1 df-nel 2402 . 2  |-  ( B  e/  C  <->  -.  B  e.  C )
2 nelne2 2397 . 2  |-  ( ( A  e.  C  /\  -.  B  e.  C
)  ->  A  =/=  B )
31, 2sylan2b 285 1  |-  ( ( A  e.  C  /\  B  e/  C )  ->  A  =/=  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    e. wcel 1480    =/= wne 2306    e/ wnel 2401
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 603  ax-in2 604  ax-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487  ax-17 1506  ax-ial 1514  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-cleq 2130  df-clel 2133  df-ne 2307  df-nel 2402
This theorem is referenced by: (None)
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