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Theorem brdifun 6422
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 4539 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X )  -> 
<. A ,  B >.  e.  ( X  X.  X
) )
2 df-br 3898 . . . 4  |-  ( A ( X  X.  X
) B  <->  <. A ,  B >.  e.  ( X  X.  X ) )
31, 2sylibr 133 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  A ( X  X.  X ) B )
4 swoer.1 . . . . . 6  |-  R  =  ( ( X  X.  X )  \  (  .<  u.  `'  .<  )
)
54breqi 3903 . . . . 5  |-  ( A R B  <->  A (
( X  X.  X
)  \  (  .<  u.  `'  .<  ) ) B )
6 brdif 3949 . . . . 5  |-  ( A ( ( X  X.  X )  \  (  .<  u.  `'  .<  )
) B  <->  ( A
( X  X.  X
) B  /\  -.  A (  .<  u.  `'  .<  ) B ) )
75, 6bitri 183 . . . 4  |-  ( A R B  <->  ( A
( X  X.  X
) B  /\  -.  A (  .<  u.  `'  .<  ) B ) )
87baib 887 . . 3  |-  ( A ( X  X.  X
) B  ->  ( A R B  <->  -.  A
(  .<  u.  `'  .<  ) B ) )
93, 8syl 14 . 2  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A R B  <->  -.  A (  .<  u.  `'  .<  ) B ) )
10 brun 3947 . . . 4  |-  ( A (  .<  u.  `'  .<  ) B  <->  ( A  .<  B  \/  A `'  .<  B ) )
11 brcnvg 4688 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A `'  .<  B  <-> 
B  .<  A ) )
1211orbi2d 762 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( ( A  .<  B  \/  A `'  .<  B )  <->  ( A  .<  B  \/  B  .<  A ) ) )
1310, 12syl5bb 191 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A (  .<  u.  `'  .<  ) B  <->  ( A  .<  B  \/  B  .<  A ) ) )
1413notbid 639 . 2  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( -.  A ( 
.<  u.  `'  .<  ) B 
<->  -.  ( A  .<  B  \/  B  .<  A ) ) )
159, 14bitrd 187 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    /\ wa 103    <-> wb 104    \/ wo 680    = wceq 1314    e. wcel 1463    \ cdif 3036    u. cun 3037   <.cop 3498   class class class wbr 3897    X. cxp 4505   `'ccnv 4506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-br 3898  df-opab 3958  df-xp 4513  df-cnv 4515
This theorem is referenced by:  swoer  6423  swoord1  6424  swoord2  6425
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