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Theorem prtlem80 26827
Description: Lemma for prter2 26852. (Contributed by Rodolfo Medina, 17-Oct-2010.)
Assertion
Ref Expression
prtlem80  |-  ( A  e.  B  ->  -.  A  e.  ( C  \  { A } ) )

Proof of Theorem prtlem80
StepHypRef Expression
1 snidg 3678 . . . 4  |-  ( A  e.  B  ->  A  e.  { A } )
21a1d 22 . . 3  |-  ( A  e.  B  ->  ( A  e.  C  ->  A  e.  { A }
) )
3 iman 413 . . 3  |-  ( ( A  e.  C  ->  A  e.  { A } )  <->  -.  ( A  e.  C  /\  -.  A  e.  { A } ) )
42, 3sylib 188 . 2  |-  ( A  e.  B  ->  -.  ( A  e.  C  /\  -.  A  e.  { A } ) )
5 eldif 3175 . 2  |-  ( A  e.  ( C  \  { A } )  <->  ( A  e.  C  /\  -.  A  e.  { A } ) )
64, 5sylnibr 296 1  |-  ( A  e.  B  ->  -.  A  e.  ( C  \  { A } ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    e. wcel 1696    \ cdif 3162   {csn 3653
This theorem is referenced by:  prter2  26852
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  df-sn 3659
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