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Theorem brdifun 6687
Description: Evaluate the incomparability relation. (Contributed by Mario Carneiro, 9-Jul-2014.)
Hypothesis
Ref Expression
swoer.1  |-  R  =  ( ( X  X.  X )  \  (  .<  u.  `'  .<  )
)
Assertion
Ref Expression
brdifun  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A R B  <->  -.  ( A  .<  B  \/  B  .<  A ) ) )

Proof of Theorem brdifun
StepHypRef Expression
1 opelxpi 4721 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X )  -> 
<. A ,  B >.  e.  ( X  X.  X
) )
2 df-br 4024 . . . 4  |-  ( A ( X  X.  X
) B  <->  <. A ,  B >.  e.  ( X  X.  X ) )
31, 2sylibr 203 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  A ( X  X.  X ) B )
4 swoer.1 . . . . . 6  |-  R  =  ( ( X  X.  X )  \  (  .<  u.  `'  .<  )
)
54breqi 4029 . . . . 5  |-  ( A R B  <->  A (
( X  X.  X
)  \  (  .<  u.  `'  .<  ) ) B )
6 brdif 4071 . . . . 5  |-  ( A ( ( X  X.  X )  \  (  .<  u.  `'  .<  )
) B  <->  ( A
( X  X.  X
) B  /\  -.  A (  .<  u.  `'  .<  ) B ) )
75, 6bitri 240 . . . 4  |-  ( A R B  <->  ( A
( X  X.  X
) B  /\  -.  A (  .<  u.  `'  .<  ) B ) )
87baib 871 . . 3  |-  ( A ( X  X.  X
) B  ->  ( A R B  <->  -.  A
(  .<  u.  `'  .<  ) B ) )
93, 8syl 15 . 2  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A R B  <->  -.  A (  .<  u.  `'  .<  ) B ) )
10 brun 4069 . . . 4  |-  ( A (  .<  u.  `'  .<  ) B  <->  ( A  .<  B  \/  A `'  .<  B ) )
11 brcnvg 4862 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A `'  .<  B  <-> 
B  .<  A ) )
1211orbi2d 682 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( ( A  .<  B  \/  A `'  .<  B )  <->  ( A  .<  B  \/  B  .<  A ) ) )
1310, 12syl5bb 248 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A (  .<  u.  `'  .<  ) B  <->  ( A  .<  B  \/  B  .<  A ) ) )
1413notbid 285 . 2  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( -.  A ( 
.<  u.  `'  .<  ) B 
<->  -.  ( A  .<  B  \/  B  .<  A ) ) )
159, 14bitrd 244 1  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A R B  <->  -.  ( A  .<  B  \/  B  .<  A ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    = wceq 1623    e. wcel 1684    \ cdif 3149    u. cun 3150   <.cop 3643   class class class wbr 4023    X. cxp 4687   `'ccnv 4688
This theorem is referenced by:  swoer  6688  swoord1  6689  swoord2  6690  swoso  6691
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-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-cnv 4697
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