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Theorem prtlem90 30526
Description: Lemma for prter2 30550. (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 384 . . . . . . 7  |-  ( ( -.  A  e.  B  /\  C  e.  B
)  ->  ( ( A  e.  B  /\  -.  C  e.  B
)  \/  ( -.  A  e.  B  /\  C  e.  B )
) )
2 ancom 450 . . . . . . . 8  |-  ( ( -.  A  e.  B  /\  C  e.  B
)  <->  ( C  e.  B  /\  -.  A  e.  B ) )
32orbi2i 519 . . . . . . 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 196 . . . . . 6  |-  ( ( -.  A  e.  B  /\  C  e.  B
)  ->  ( ( A  e.  B  /\  -.  C  e.  B
)  \/  ( C  e.  B  /\  -.  A  e.  B )
) )
5 xor 889 . . . . . 6  |-  ( -.  ( A  e.  B  <->  C  e.  B )  <->  ( ( A  e.  B  /\  -.  C  e.  B
)  \/  ( C  e.  B  /\  -.  A  e.  B )
) )
64, 5sylibr 212 . . . . 5  |-  ( ( -.  A  e.  B  /\  C  e.  B
)  ->  -.  ( A  e.  B  <->  C  e.  B ) )
7 eleq1 2539 . . . . 5  |-  ( A  =  C  ->  ( A  e.  B  <->  C  e.  B ) )
86, 7nsyl 121 . . . 4  |-  ( ( -.  A  e.  B  /\  C  e.  B
)  ->  -.  A  =  C )
98ex 434 . . 3  |-  ( -.  A  e.  B  -> 
( C  e.  B  ->  -.  A  =  C ) )
10 df-ne 2664 . . 3  |-  ( A  =/=  C  <->  -.  A  =  C )
119, 10syl6ibr 227 . 2  |-  ( -.  A  e.  B  -> 
( C  e.  B  ->  A  =/=  C ) )
12 necom 2736 . 2  |-  ( A  =/=  C  <->  C  =/=  A )
1311, 12syl6ib 226 1  |-  ( -.  A  e.  B  -> 
( C  e.  B  ->  C  =/=  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1597  df-cleq 2459  df-clel 2462  df-ne 2664
This theorem is referenced by:  prter2  30550
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