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Theorem nelneq2 2574
Description: A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)
Assertion
Ref Expression
nelneq2  |-  ( ( A  e.  B  /\  -.  A  e.  C
)  ->  -.  B  =  C )

Proof of Theorem nelneq2
StepHypRef Expression
1 eleq2 2538 . . 3  |-  ( B  =  C  ->  ( A  e.  B  <->  A  e.  C ) )
21biimpcd 232 . 2  |-  ( A  e.  B  ->  ( B  =  C  ->  A  e.  C ) )
32con3dimp 448 1  |-  ( ( A  e.  B  /\  -.  A  e.  C
)  ->  -.  B  =  C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-cleq 2464  df-clel 2467
This theorem is referenced by:  ssnelpss  3530  opthwiener  4703  ssfin4  8758  pwxpndom2  9108  fzneuz  11901  hauspwpwf1  21080  vdgr1b  25711  topdifinffinlem  31820
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