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Theorem eltpg 3576
Description: Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
eltpg  |-  ( A  e.  V  ->  ( A  e.  { B ,  C ,  D }  <->  ( A  =  B  \/  A  =  C  \/  A  =  D )
) )

Proof of Theorem eltpg
StepHypRef Expression
1 elprg 3552 . . 3  |-  ( A  e.  V  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) ) )
2 elsng 3547 . . 3  |-  ( A  e.  V  ->  ( A  e.  { D } 
<->  A  =  D ) )
31, 2orbi12d 783 . 2  |-  ( A  e.  V  ->  (
( A  e.  { B ,  C }  \/  A  e.  { D } )  <->  ( ( A  =  B  \/  A  =  C )  \/  A  =  D
) ) )
4 df-tp 3540 . . . 4  |-  { B ,  C ,  D }  =  ( { B ,  C }  u.  { D } )
54eleq2i 2207 . . 3  |-  ( A  e.  { B ,  C ,  D }  <->  A  e.  ( { B ,  C }  u.  { D } ) )
6 elun 3222 . . 3  |-  ( A  e.  ( { B ,  C }  u.  { D } )  <->  ( A  e.  { B ,  C }  \/  A  e.  { D } ) )
75, 6bitri 183 . 2  |-  ( A  e.  { B ,  C ,  D }  <->  ( A  e.  { B ,  C }  \/  A  e.  { D } ) )
8 df-3or 964 . 2  |-  ( ( A  =  B  \/  A  =  C  \/  A  =  D )  <->  ( ( A  =  B  \/  A  =  C )  \/  A  =  D ) )
93, 7, 83bitr4g 222 1  |-  ( A  e.  V  ->  ( A  e.  { B ,  C ,  D }  <->  ( A  =  B  \/  A  =  C  \/  A  =  D )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    \/ wo 698    \/ w3o 962    = wceq 1332    e. wcel 1481    u. cun 3074   {csn 3532   {cpr 3533   {ctp 3534
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-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3or 964  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3080  df-sn 3538  df-pr 3539  df-tp 3540
This theorem is referenced by:  eldiftp  3577  eltpi  3578  eltp  3579  tpid1g  3643  tpid2g  3645
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