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Theorem nelneq2 2540
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 2502 . . 3  |-  ( B  =  C  ->  ( A  e.  B  <->  A  e.  C ) )
21biimpcd 224 . 2  |-  ( A  e.  B  ->  ( B  =  C  ->  A  e.  C ) )
32con3and 439 1  |-  ( ( A  e.  B  /\  -.  A  e.  C
)  ->  -.  B  =  C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1364    e. wcel 1761
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-12 1797  ax-ext 2422
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1592  df-cleq 2434  df-clel 2437
This theorem is referenced by:  ssnelpss  3739  opthwiener  4590  ssfin4  8475  pwxpndom2  8828  fzneuz  11537  hauspwpwf1  19460  vdgr1b  23493
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