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Theorem nelneq2 2395
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 2357 . . 3  |-  ( B  =  C  ->  ( A  e.  B  <->  A  e.  C ) )
21biimpcd 215 . 2  |-  ( A  e.  B  ->  ( B  =  C  ->  A  e.  C ) )
32con3and 428 1  |-  ( ( A  e.  B  /\  -.  A  e.  C
)  ->  -.  B  =  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696
This theorem is referenced by:  ssnelpss  3530  opthwiener  4284  ssfin4  7952  pwxpndom2  8303  fzneuz  10879  hauspwpwf1  17698  vdgr1b  23910
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-cleq 2289  df-clel 2292
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