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Theorem brdifun 6924
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 4902 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X )  -> 
<. A ,  B >.  e.  ( X  X.  X
) )
2 df-br 4205 . . . 4  |-  ( A ( X  X.  X
) B  <->  <. A ,  B >.  e.  ( X  X.  X ) )
31, 2sylibr 204 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  A ( X  X.  X ) B )
4 swoer.1 . . . . . 6  |-  R  =  ( ( X  X.  X )  \  (  .<  u.  `'  .<  )
)
54breqi 4210 . . . . 5  |-  ( A R B  <->  A (
( X  X.  X
)  \  (  .<  u.  `'  .<  ) ) B )
6 brdif 4252 . . . . 5  |-  ( A ( ( X  X.  X )  \  (  .<  u.  `'  .<  )
) B  <->  ( A
( X  X.  X
) B  /\  -.  A (  .<  u.  `'  .<  ) B ) )
75, 6bitri 241 . . . 4  |-  ( A R B  <->  ( A
( X  X.  X
) B  /\  -.  A (  .<  u.  `'  .<  ) B ) )
87baib 872 . . 3  |-  ( A ( X  X.  X
) B  ->  ( A R B  <->  -.  A
(  .<  u.  `'  .<  ) B ) )
93, 8syl 16 . 2  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A R B  <->  -.  A (  .<  u.  `'  .<  ) B ) )
10 brun 4250 . . . 4  |-  ( A (  .<  u.  `'  .<  ) B  <->  ( A  .<  B  \/  A `'  .<  B ) )
11 brcnvg 5045 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A `'  .<  B  <-> 
B  .<  A ) )
1211orbi2d 683 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( ( A  .<  B  \/  A `'  .<  B )  <->  ( A  .<  B  \/  B  .<  A ) ) )
1310, 12syl5bb 249 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A (  .<  u.  `'  .<  ) B  <->  ( A  .<  B  \/  B  .<  A ) ) )
1413notbid 286 . 2  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( -.  A ( 
.<  u.  `'  .<  ) B 
<->  -.  ( A  .<  B  \/  B  .<  A ) ) )
159, 14bitrd 245 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 177    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725    \ cdif 3309    u. cun 3310   <.cop 3809   class class class wbr 4204    X. cxp 4868   `'ccnv 4869
This theorem is referenced by:  swoer  6925  swoord1  6926  swoord2  6927  swoso  6928
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-cnv 4878
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