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

Step	Hyp	Ref	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 ) )
3	1, 2	orbi12d 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 } )
5	4	eleq2i 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 } ) )
7	5, 6	bitri 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 ) )
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 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