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Theorem eltpg 3516
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 3494 . . 3  |-  ( A  e.  V  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) ) )
2 elsng 3489 . . 3  |-  ( A  e.  V  ->  ( A  e.  { D } 
<->  A  =  D ) )
31, 2orbi12d 748 . 2  |-  ( A  e.  V  ->  (
( A  e.  { B ,  C }  \/  A  e.  { D } )  <->  ( ( A  =  B  \/  A  =  C )  \/  A  =  D
) ) )
4 df-tp 3482 . . . 4  |-  { B ,  C ,  D }  =  ( { B ,  C }  u.  { D } )
54eleq2i 2166 . . 3  |-  ( A  e.  { B ,  C ,  D }  <->  A  e.  ( { B ,  C }  u.  { D } ) )
6 elun 3164 . . 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 931 . 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 670    \/ w3o 929    = wceq 1299    e. wcel 1448    u. cun 3019   {csn 3474   {cpr 3475   {ctp 3476
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-3or 931  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-sn 3480  df-pr 3481  df-tp 3482
This theorem is referenced by:  eltpi  3517  eltp  3518  tpid1g  3582  tpid2g  3584
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