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Theorem nelneq 2519
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 2474 . . 3  |-  ( A  =  B  ->  ( A  e.  C  <->  B  e.  C ) )
21biimpcd 224 . 2  |-  ( A  e.  C  ->  ( A  =  B  ->  B  e.  C ) )
32con3dimp 439 1  |-  ( ( A  e.  C  /\  -.  B  e.  C
)  ->  -.  A  =  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    = wceq 1405    e. wcel 1842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-ex 1634  df-cleq 2394  df-clel 2397
This theorem is referenced by:  onfununi  7044  suc11reg  8068  cantnfp1lem3  8130  oemapvali  8134  cantnfp1lem3OLD  8156  mreexmrid  15255  supxrnemnf  28017  xrge0neqmnf  28117  onint1  30668  maxidln0  31704  rencldnfilem  35095  icccncfext  37039
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