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Theorem nelneq 2573
Description: A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)
Assertion
Ref Expression
nelneq  |-  ( ( A  e.  C  /\  -.  B  e.  C
)  ->  -.  A  =  B )

Proof of Theorem nelneq
StepHypRef Expression
1 eleq1 2537 . . 3  |-  ( A  =  B  ->  ( A  e.  C  <->  B  e.  C ) )
21biimpcd 232 . 2  |-  ( A  e.  C  ->  ( A  =  B  ->  B  e.  C ) )
32con3dimp 448 1  |-  ( ( A  e.  C  /\  -.  B  e.  C
)  ->  -.  A  =  B )
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-5 1766  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:  onfununi  7078  suc11reg  8142  cantnfp1lem3  8203  oemapvali  8207  xrge0neqmnf  11762  mreexmrid  15627  supxrnemnf  28429  onint1  31180  maxidln0  32342  rencldnfilem  35734  icccncfext  37862
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