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Theorem neldif 3314
Description: Implication of membership in a class difference. (Contributed by NM, 28-Jun-1994.)
Assertion
Ref Expression
neldif  |-  ( ( A  e.  B  /\  -.  A  e.  ( B  \  C ) )  ->  A  e.  C
)

Proof of Theorem neldif
StepHypRef Expression
1 eldif 3175 . . . 4  |-  ( A  e.  ( B  \  C )  <->  ( A  e.  B  /\  -.  A  e.  C ) )
21simplbi2 608 . . 3  |-  ( A  e.  B  ->  ( -.  A  e.  C  ->  A  e.  ( B 
\  C ) ) )
32con1d 116 . 2  |-  ( A  e.  B  ->  ( -.  A  e.  ( B  \  C )  ->  A  e.  C )
)
43imp 418 1  |-  ( ( A  e.  B  /\  -.  A  e.  ( B  \  C ) )  ->  A  e.  C
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    e. wcel 1696    \ cdif 3162
This theorem is referenced by:  peano5  4695  boxcutc  6875
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-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-dif 3168
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