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Theorem prtlem90 26708
Description: Lemma for prter2 26732. (Contributed by Rodolfo Medina, 17-Oct-2010.)
Assertion
Ref Expression
prtlem90  |-  ( -.  A  e.  B  -> 
( C  e.  B  ->  C  =/=  A ) )

Proof of Theorem prtlem90
StepHypRef Expression
1 olc 375 . . . . . . 7  |-  ( ( -.  A  e.  B  /\  C  e.  B
)  ->  ( ( A  e.  B  /\  -.  C  e.  B
)  \/  ( -.  A  e.  B  /\  C  e.  B )
) )
2 ancom 439 . . . . . . . 8  |-  ( ( -.  A  e.  B  /\  C  e.  B
)  <->  ( C  e.  B  /\  -.  A  e.  B ) )
32orbi2i 507 . . . . . . 7  |-  ( ( ( A  e.  B  /\  -.  C  e.  B
)  \/  ( -.  A  e.  B  /\  C  e.  B )
)  <->  ( ( A  e.  B  /\  -.  C  e.  B )  \/  ( C  e.  B  /\  -.  A  e.  B
) ) )
41, 3sylib 190 . . . . . 6  |-  ( ( -.  A  e.  B  /\  C  e.  B
)  ->  ( ( A  e.  B  /\  -.  C  e.  B
)  \/  ( C  e.  B  /\  -.  A  e.  B )
) )
5 xor 863 . . . . . 6  |-  ( -.  ( A  e.  B  <->  C  e.  B )  <->  ( ( A  e.  B  /\  -.  C  e.  B
)  \/  ( C  e.  B  /\  -.  A  e.  B )
) )
64, 5sylibr 205 . . . . 5  |-  ( ( -.  A  e.  B  /\  C  e.  B
)  ->  -.  ( A  e.  B  <->  C  e.  B ) )
7 eleq1 2498 . . . . 5  |-  ( A  =  C  ->  ( A  e.  B  <->  C  e.  B ) )
86, 7nsyl 116 . . . 4  |-  ( ( -.  A  e.  B  /\  C  e.  B
)  ->  -.  A  =  C )
98ex 425 . . 3  |-  ( -.  A  e.  B  -> 
( C  e.  B  ->  -.  A  =  C ) )
10 df-ne 2603 . . 3  |-  ( A  =/=  C  <->  -.  A  =  C )
119, 10syl6ibr 220 . 2  |-  ( -.  A  e.  B  -> 
( C  e.  B  ->  A  =/=  C ) )
12 necom 2687 . 2  |-  ( A  =/=  C  <->  C  =/=  A )
1311, 12syl6ib 219 1  |-  ( -.  A  e.  B  -> 
( C  e.  B  ->  C  =/=  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1653    e. wcel 1726    =/= wne 2601
This theorem is referenced by:  prter2  26732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-11 1762  ax-ext 2419
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-ex 1552  df-cleq 2431  df-clel 2434  df-ne 2603
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