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Theorem prtlem90 25871
Description: Lemma for prter2 25897. (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 373 . . . . . . 7  |-  ( ( -.  A  e.  B  /\  C  e.  B
)  ->  ( ( A  e.  B  /\  -.  C  e.  B
)  \/  ( -.  A  e.  B  /\  C  e.  B )
) )
2 ancom 437 . . . . . . . 8  |-  ( ( -.  A  e.  B  /\  C  e.  B
)  <->  ( C  e.  B  /\  -.  A  e.  B ) )
32orbi2i 505 . . . . . . 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 188 . . . . . 6  |-  ( ( -.  A  e.  B  /\  C  e.  B
)  ->  ( ( A  e.  B  /\  -.  C  e.  B
)  \/  ( C  e.  B  /\  -.  A  e.  B )
) )
5 xor 861 . . . . . 6  |-  ( -.  ( A  e.  B  <->  C  e.  B )  <->  ( ( A  e.  B  /\  -.  C  e.  B
)  \/  ( C  e.  B  /\  -.  A  e.  B )
) )
64, 5sylibr 203 . . . . 5  |-  ( ( -.  A  e.  B  /\  C  e.  B
)  ->  -.  ( A  e.  B  <->  C  e.  B ) )
7 eleq1 2376 . . . . 5  |-  ( A  =  C  ->  ( A  e.  B  <->  C  e.  B ) )
86, 7nsyl 113 . . . 4  |-  ( ( -.  A  e.  B  /\  C  e.  B
)  ->  -.  A  =  C )
98ex 423 . . 3  |-  ( -.  A  e.  B  -> 
( C  e.  B  ->  -.  A  =  C ) )
10 df-ne 2481 . . 3  |-  ( A  =/=  C  <->  -.  A  =  C )
119, 10syl6ibr 218 . 2  |-  ( -.  A  e.  B  -> 
( C  e.  B  ->  A  =/=  C ) )
12 necom 2560 . 2  |-  ( A  =/=  C  <->  C  =/=  A )
1311, 12syl6ib 217 1  |-  ( -.  A  e.  B  -> 
( C  e.  B  ->  C  =/=  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1633    e. wcel 1701    =/= wne 2479
This theorem is referenced by:  prter2  25897
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-11 1732  ax-ext 2297
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-ex 1533  df-cleq 2309  df-clel 2312  df-ne 2481
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