ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  brdifun Unicode version

Theorem brdifun 6540
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 4643 . . . 4  |-  ( ( A  e.  X  /\  B  e.  X )  -> 
<. A ,  B >.  e.  ( X  X.  X
) )
2 df-br 3990 . . . 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 3995 . . . . 5  |-  ( A R B  <->  A (
( X  X.  X
)  \  (  .<  u.  `'  .<  ) ) B )
6 brdif 4042 . . . . 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 914 . . 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 4040 . . . 4  |-  ( A (  .<  u.  `'  .<  ) B  <->  ( A  .<  B  \/  A `'  .<  B ) )
11 brcnvg 4792 . . . . 5  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A `'  .<  B  <-> 
B  .<  A ) )
1211orbi2d 785 . . . 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 662 . 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 703    = wceq 1348    e. wcel 2141    \ cdif 3118    u. cun 3119   <.cop 3586   class class class wbr 3989    X. cxp 4609   `'ccnv 4610
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-xp 4617  df-cnv 4619
This theorem is referenced by:  swoer  6541  swoord1  6542  swoord2  6543
  Copyright terms: Public domain W3C validator