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Theorem brtp 23478
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 3998 . 2  |-  ( X { <. A ,  B >. ,  <. C ,  D >. ,  <. E ,  F >. } Y  <->  <. X ,  Y >.  e.  { <. A ,  B >. ,  <. C ,  D >. ,  <. E ,  F >. } )
2 opex 4209 . . 3  |-  <. X ,  Y >.  e.  _V
32eltp 3652 . 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 4217 . . 3  |-  ( <. X ,  Y >.  = 
<. A ,  B >.  <->  ( X  =  A  /\  Y  =  B )
)
74, 5opth 4217 . . 3  |-  ( <. X ,  Y >.  = 
<. C ,  D >.  <->  ( X  =  C  /\  Y  =  D )
)
84, 5opth 4217 . . 3  |-  ( <. X ,  Y >.  = 
<. E ,  F >.  <->  ( X  =  E  /\  Y  =  F )
)
96, 7, 83orbi123i 1146 . 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 264 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 178    /\ wa 360    \/ w3o 938    = wceq 1619    e. wcel 1621   _Vcvv 2763   {ctp 3616   <.cop 3617   class class class wbr 3997
This theorem is referenced by:  sltval2  23679  sltsgn1  23684  sltsgn2  23685  sltintdifex  23686  sltres  23687  axsltsolem1  23691  axdenselem8  23712  axdense  23713  axfelem9  23724  axfelem10  23725
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-rab 2527  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-tp 3622  df-op 3623  df-br 3998
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