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Theorem eltpg 3444
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 3423 . . 3  |-  ( A  e.  V  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) ) )
2 elsng 3418 . . 3  |-  ( A  e.  V  ->  ( A  e.  { D } 
<->  A  =  D ) )
31, 2orbi12d 717 . 2  |-  ( A  e.  V  ->  (
( A  e.  { B ,  C }  \/  A  e.  { D } )  <->  ( ( A  =  B  \/  A  =  C )  \/  A  =  D
) ) )
4 df-tp 3411 . . . 4  |-  { B ,  C ,  D }  =  ( { B ,  C }  u.  { D } )
54eleq2i 2120 . . 3  |-  ( A  e.  { B ,  C ,  D }  <->  A  e.  ( { B ,  C }  u.  { D } ) )
6 elun 3112 . . 3  |-  ( A  e.  ( { B ,  C }  u.  { D } )  <->  ( A  e.  { B ,  C }  \/  A  e.  { D } ) )
75, 6bitri 177 . 2  |-  ( A  e.  { B ,  C ,  D }  <->  ( A  e.  { B ,  C }  \/  A  e.  { D } ) )
8 df-3or 897 . 2  |-  ( ( A  =  B  \/  A  =  C  \/  A  =  D )  <->  ( ( A  =  B  \/  A  =  C )  \/  A  =  D ) )
93, 7, 83bitr4g 216 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 102    \/ wo 639    \/ w3o 895    = wceq 1259    e. wcel 1409    u. cun 2943   {csn 3403   {cpr 3404   {ctp 3405
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3or 897  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2950  df-sn 3409  df-pr 3410  df-tp 3411
This theorem is referenced by:  eltpi  3445  eltp  3446
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