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Theorem prtlem90 32340
Description: Lemma for prter2 32364. (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 385 . . . . . . 7  |-  ( ( -.  A  e.  B  /\  C  e.  B
)  ->  ( ( A  e.  B  /\  -.  C  e.  B
)  \/  ( -.  A  e.  B  /\  C  e.  B )
) )
2 ancom 451 . . . . . . . 8  |-  ( ( -.  A  e.  B  /\  C  e.  B
)  <->  ( C  e.  B  /\  -.  A  e.  B ) )
32orbi2i 521 . . . . . . 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 199 . . . . . 6  |-  ( ( -.  A  e.  B  /\  C  e.  B
)  ->  ( ( A  e.  B  /\  -.  C  e.  B
)  \/  ( C  e.  B  /\  -.  A  e.  B )
) )
5 xor 899 . . . . . 6  |-  ( -.  ( A  e.  B  <->  C  e.  B )  <->  ( ( A  e.  B  /\  -.  C  e.  B
)  \/  ( C  e.  B  /\  -.  A  e.  B )
) )
64, 5sylibr 215 . . . . 5  |-  ( ( -.  A  e.  B  /\  C  e.  B
)  ->  -.  ( A  e.  B  <->  C  e.  B ) )
7 eleq1 2494 . . . . 5  |-  ( A  =  C  ->  ( A  e.  B  <->  C  e.  B ) )
86, 7nsyl 124 . . . 4  |-  ( ( -.  A  e.  B  /\  C  e.  B
)  ->  -.  A  =  C )
98ex 435 . . 3  |-  ( -.  A  e.  B  -> 
( C  e.  B  ->  -.  A  =  C ) )
10 df-ne 2601 . . 3  |-  ( A  =/=  C  <->  -.  A  =  C )
119, 10syl6ibr 230 . 2  |-  ( -.  A  e.  B  -> 
( C  e.  B  ->  A  =/=  C ) )
12 necom 2654 . 2  |-  ( A  =/=  C  <->  C  =/=  A )
1311, 12syl6ib 229 1  |-  ( -.  A  e.  B  -> 
( C  e.  B  ->  C  =/=  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 187    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872    =/= wne 2599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-ex 1658  df-cleq 2421  df-clel 2424  df-ne 2601
This theorem is referenced by:  prter2  32364
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