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Theorem brtp 24664
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 4103 . 2  |-  ( X { <. A ,  B >. ,  <. C ,  D >. ,  <. E ,  F >. } Y  <->  <. X ,  Y >.  e.  { <. A ,  B >. ,  <. C ,  D >. ,  <. E ,  F >. } )
2 opex 4316 . . 3  |-  <. X ,  Y >.  e.  _V
32eltp 3754 . 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 4324 . . 3  |-  ( <. X ,  Y >.  = 
<. A ,  B >.  <->  ( X  =  A  /\  Y  =  B )
)
74, 5opth 4324 . . 3  |-  ( <. X ,  Y >.  = 
<. C ,  D >.  <->  ( X  =  C  /\  Y  =  D )
)
84, 5opth 4324 . . 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 1642    e. wcel 1710   _Vcvv 2864   {ctp 3718   <.cop 3719   class class class wbr 4102
This theorem is referenced by:  sltval2  24868  sltsgn1  24873  sltsgn2  24874  sltintdifex  24875  sltres  24876  sltsolem1  24880  nodenselem8  24900  nodense  24901  nobndup  24912  nobnddown  24913
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-rab 2628  df-v 2866  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-tp 3724  df-op 3725  df-br 4103
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