Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  brtp Unicode version

Theorem brtp 24106
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 4024 . 2  |-  ( X { <. A ,  B >. ,  <. C ,  D >. ,  <. E ,  F >. } Y  <->  <. X ,  Y >.  e.  { <. A ,  B >. ,  <. C ,  D >. ,  <. E ,  F >. } )
2 opex 4237 . . 3  |-  <. X ,  Y >.  e.  _V
32eltp 3678 . 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 4245 . . 3  |-  ( <. X ,  Y >.  = 
<. A ,  B >.  <->  ( X  =  A  /\  Y  =  B )
)
74, 5opth 4245 . . 3  |-  ( <. X ,  Y >.  = 
<. C ,  D >.  <->  ( X  =  C  /\  Y  =  D )
)
84, 5opth 4245 . . 3  |-  ( <. X ,  Y >.  = 
<. E ,  F >.  <->  ( X  =  E  /\  Y  =  F )
)
96, 7, 83orbi123i 1141 . 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 262 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 176    /\ wa 358    \/ w3o 933    = wceq 1623    e. wcel 1684   _Vcvv 2788   {ctp 3642   <.cop 3643   class class class wbr 4023
This theorem is referenced by:  sltval2  24310  sltsgn1  24315  sltsgn2  24316  sltintdifex  24317  sltres  24318  sltsolem1  24322  nodenselem8  24342  nodense  24343  nobndup  24354  nobnddown  24355
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-br 4024
  Copyright terms: Public domain W3C validator