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Theorem nelneq 2394
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 2356 . . 3  |-  ( A  =  B  ->  ( A  e.  C  <->  B  e.  C ) )
21biimpcd 215 . 2  |-  ( A  e.  C  ->  ( A  =  B  ->  B  e.  C ) )
32con3and 428 1  |-  ( ( A  e.  C  /\  -.  B  e.  C
)  ->  -.  A  =  B )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696
This theorem is referenced by:  onfununi  6374  suc11reg  7336  cantnfp1lem3  7398  oemapvali  7402  mreexmrid  13561  supxrnemnf  23271  xrge0neqmnf  23345  onint1  24960  lppotos  26247  maxidln0  26773  rencldnfilem  27006
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-11 1727  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1532  df-cleq 2289  df-clel 2292
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