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Theorem nelneq 2540
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 2495 . . 3  |-  ( A  =  B  ->  ( A  e.  C  <->  B  e.  C ) )
21biimpcd 228 . 2  |-  ( A  e.  C  ->  ( A  =  B  ->  B  e.  C ) )
32con3dimp 443 1  |-  ( ( A  e.  C  /\  -.  B  e.  C
)  ->  -.  A  =  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    = wceq 1438    e. wcel 1869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1661  df-cleq 2415  df-clel 2418
This theorem is referenced by:  onfununi  7066  suc11reg  8128  cantnfp1lem3  8188  oemapvali  8192  xrge0neqmnf  11739  mreexmrid  15542  supxrnemnf  28354  onint1  31108  maxidln0  32236  rencldnfilem  35626  icccncfext  37629
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