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Theorem brdifun 6586
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 4676 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X )  -> 
<. A ,  B >.  e.  ( X  X.  X
) )
2 df-br 4019 . . . 4  |-  ( A ( X  X.  X
) B  <->  <. A ,  B >.  e.  ( X  X.  X ) )
31, 2sylibr 134 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  A ( X  X.  X ) B )
4 swoer.1 . . . . . 6  |-  R  =  ( ( X  X.  X )  \  (  .<  u.  `'  .<  )
)
54breqi 4024 . . . . 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 184 . . . 4  |-  ( A R B  <->  ( A
( X  X.  X
) B  /\  -.  A (  .<  u.  `'  .<  ) B ) )
87baib 920 . . 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 4069 . . . 4  |-  ( A (  .<  u.  `'  .<  ) B  <->  ( A  .<  B  \/  A `'  .<  B ) )
11 brcnvg 4826 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A `'  .<  B  <-> 
B  .<  A ) )
1211orbi2d 791 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( ( A  .<  B  \/  A `'  .<  B )  <->  ( A  .<  B  \/  B  .<  A ) ) )
1310, 12bitrid 192 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A (  .<  u.  `'  .<  ) B  <->  ( A  .<  B  \/  B  .<  A ) ) )
1413notbid 668 . 2  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( -.  A ( 
.<  u.  `'  .<  ) B 
<->  -.  ( A  .<  B  \/  B  .<  A ) ) )
159, 14bitrd 188 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 104    <-> wb 105    \/ wo 709    = wceq 1364    e. wcel 2160    \ cdif 3141    u. cun 3142   <.cop 3610   class class class wbr 4018    X. cxp 4642   `'ccnv 4643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-xp 4650  df-cnv 4652
This theorem is referenced by:  swoer  6587  swoord1  6588  swoord2  6589
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