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Theorem neldif 3472
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 3330 . . . 4  |-  ( A  e.  ( B  \  C )  <->  ( A  e.  B  /\  -.  A  e.  C ) )
21simplbi2 609 . . 3  |-  ( A  e.  B  ->  ( -.  A  e.  C  ->  A  e.  ( B 
\  C ) ) )
32con1d 118 . 2  |-  ( A  e.  B  ->  ( -.  A  e.  ( B  \  C )  ->  A  e.  C )
)
43imp 419 1  |-  ( ( A  e.  B  /\  -.  A  e.  ( B  \  C ) )  ->  A  e.  C
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 359    e. wcel 1725    \ cdif 3317
This theorem is referenced by:  peano5  4868  boxcutc  7105
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-dif 3323
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