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Theorem nelneq2 2534
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 2496 . . 3  |-  ( B  =  C  ->  ( A  e.  B  <->  A  e.  C ) )
21biimpcd 216 . 2  |-  ( A  e.  B  ->  ( B  =  C  ->  A  e.  C ) )
32con3and 429 1  |-  ( ( A  e.  B  /\  -.  A  e.  C
)  ->  -.  B  =  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725
This theorem is referenced by:  ssnelpss  3683  opthwiener  4450  ssfin4  8182  pwxpndom2  8532  fzneuz  11120  hauspwpwf1  18011  vdgr1b  21667
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-11 1761  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-cleq 2428  df-clel 2431
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