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Theorem nelneq 2579
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 2534 . . 3  |-  ( A  =  B  ->  ( A  e.  C  <->  B  e.  C ) )
21biimpcd 224 . 2  |-  ( A  e.  C  ->  ( A  =  B  ->  B  e.  C ) )
32con3dimp 441 1  |-  ( ( A  e.  C  /\  -.  B  e.  C
)  ->  -.  A  =  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1374    e. wcel 1762
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-ext 2440
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1592  df-cleq 2454  df-clel 2457
This theorem is referenced by:  onfununi  7004  suc11reg  8027  cantnfp1lem3  8090  oemapvali  8094  cantnfp1lem3OLD  8116  mreexmrid  14889  supxrnemnf  27239  xrge0neqmnf  27329  onint1  29479  maxidln0  30034  rencldnfilem  30347  icccncfext  31183
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