Theorem eltpg 3628

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

Step	Hyp	Ref	Expression
1		elprg 3603	. . 3 |- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )
2		elsng 3598	. . 3 |- ( A e. V -> ( A e. { D } <-> A = D ) )
3	1, 2	orbi12d 788	. 2 |- ( A e. V -> ( ( A e. { B , C } \/ A e. { D } ) <-> ( ( A = B \/ A = C ) \/ A = D ) ) )
4		df-tp 3591	. . . 4 |- { B , C , D } = ( { B , C } u. { D } )
5	4	eleq2i 2237	. . 3 |- ( A e. { B , C , D } <-> A e. ( { B , C } u. { D } ) )
6		elun 3268	. . 3 |- ( A e. ( { B , C } u. { D } ) <-> ( A e. { B , C } \/ A e. { D } ) )
7	5, 6	bitri 183	. 2 |- ( A e. { B , C , D } <-> ( A e. { B , C } \/ A e. { D } ) )
8		df-3or 974	. 2 |- ( ( A = B \/ A = C \/ A = D ) <-> ( ( A = B \/ A = C ) \/ A = D ) )
9	3, 7, 8	3bitr4g 222	1 |- ( A e. V -> ( A e. { B , C , D } <-> ( A = B \/ A = C \/ A = D ) ) )

Syntax hints:   -> wi 4   <-> wb 104   \/ wo 703   \/ w3o 972   = wceq 1348   e. wcel 2141   u. cun 3119   {csn 3583   {cpr 3584   {ctp 3585

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3or 974  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-tp 3591

This theorem is referenced by:  eldiftp  3629  eltpi  3630  eltp  3631  tpid1g  3695  tpid2g  3697  zabsle1  13694