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Theorem nelneq2 2537
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 2499 . . 3  |-  ( B  =  C  ->  ( A  e.  B  <->  A  e.  C ) )
21biimpcd 217 . 2  |-  ( A  e.  B  ->  ( B  =  C  ->  A  e.  C ) )
32con3and 430 1  |-  ( ( A  e.  B  /\  -.  A  e.  C
)  ->  -.  B  =  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726
This theorem is referenced by:  ssnelpss  3693  opthwiener  4461  ssfin4  8195  pwxpndom2  8545  fzneuz  11133  hauspwpwf1  18024  vdgr1b  21680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-11 1762  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-cleq 2431  df-clel 2434
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