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Theorem prtlem80 26724
Description: Lemma for prter2 26749. (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 3665 . . . 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 3162 . 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 1684    \ cdif 3149   {csn 3640
This theorem is referenced by:  prter2  26749
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-dif 3155  df-sn 3646
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