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Theorem brtp 25130
Description: A condition for a binary relation over an unordered triple. (Contributed by Scott Fenton, 8-Jun-2011.)
Hypotheses
Ref Expression
brtp.1  |-  X  e. 
_V
brtp.2  |-  Y  e. 
_V
Assertion
Ref Expression
brtp  |-  ( X { <. A ,  B >. ,  <. C ,  D >. ,  <. E ,  F >. } Y  <->  ( ( X  =  A  /\  Y  =  B )  \/  ( X  =  C  /\  Y  =  D )  \/  ( X  =  E  /\  Y  =  F ) ) )

Proof of Theorem brtp
StepHypRef Expression
1 df-br 4154 . 2  |-  ( X { <. A ,  B >. ,  <. C ,  D >. ,  <. E ,  F >. } Y  <->  <. X ,  Y >.  e.  { <. A ,  B >. ,  <. C ,  D >. ,  <. E ,  F >. } )
2 opex 4368 . . 3  |-  <. X ,  Y >.  e.  _V
32eltp 3796 . 2  |-  ( <. X ,  Y >.  e. 
{ <. A ,  B >. ,  <. C ,  D >. ,  <. E ,  F >. }  <->  ( <. X ,  Y >.  =  <. A ,  B >.  \/  <. X ,  Y >.  =  <. C ,  D >.  \/  <. X ,  Y >.  =  <. E ,  F >. ) )
4 brtp.1 . . . 4  |-  X  e. 
_V
5 brtp.2 . . . 4  |-  Y  e. 
_V
64, 5opth 4376 . . 3  |-  ( <. X ,  Y >.  = 
<. A ,  B >.  <->  ( X  =  A  /\  Y  =  B )
)
74, 5opth 4376 . . 3  |-  ( <. X ,  Y >.  = 
<. C ,  D >.  <->  ( X  =  C  /\  Y  =  D )
)
84, 5opth 4376 . . 3  |-  ( <. X ,  Y >.  = 
<. E ,  F >.  <->  ( X  =  E  /\  Y  =  F )
)
96, 7, 83orbi123i 1143 . 2  |-  ( (
<. X ,  Y >.  = 
<. A ,  B >.  \/ 
<. X ,  Y >.  = 
<. C ,  D >.  \/ 
<. X ,  Y >.  = 
<. E ,  F >. )  <-> 
( ( X  =  A  /\  Y  =  B )  \/  ( X  =  C  /\  Y  =  D )  \/  ( X  =  E  /\  Y  =  F ) ) )
101, 3, 93bitri 263 1  |-  ( X { <. A ,  B >. ,  <. C ,  D >. ,  <. E ,  F >. } Y  <->  ( ( X  =  A  /\  Y  =  B )  \/  ( X  =  C  /\  Y  =  D )  \/  ( X  =  E  /\  Y  =  F ) ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    \/ w3o 935    = wceq 1649    e. wcel 1717   _Vcvv 2899   {ctp 3759   <.cop 3760   class class class wbr 4153
This theorem is referenced by:  sltval2  25334  sltsgn1  25339  sltsgn2  25340  sltintdifex  25341  sltres  25342  sltsolem1  25346  nodenselem8  25366  nodense  25367  nobndup  25378  nobnddown  25379
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-rab 2658  df-v 2901  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-tp 3765  df-op 3766  df-br 4154
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