Theorem eltpg 3711

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 3686	. . 3 |- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )
2		elsng 3681	. . 3 |- ( A e. V -> ( A e. { D } <-> A = D ) )
3	1, 2	orbi12d 798	. 2 |- ( A e. V -> ( ( A e. { B , C } \/ A e. { D } ) <-> ( ( A = B \/ A = C ) \/ A = D ) ) )
4		df-tp 3674	. . . 4 |- { B , C , D } = ( { B , C } u. { D } )
5	4	eleq2i 2296	. . 3 |- ( A e. { B , C , D } <-> A e. ( { B , C } u. { D } ) )
6		elun 3345	. . 3 |- ( A e. ( { B , C } u. { D } ) <-> ( A e. { B , C } \/ A e. { D } ) )
7	5, 6	bitri 184	. 2 |- ( A e. { B , C , D } <-> ( A e. { B , C } \/ A e. { D } ) )
8		df-3or 1003	. 2 |- ( ( A = B \/ A = C \/ A = D ) <-> ( ( A = B \/ A = C ) \/ A = D ) )
9	3, 7, 8	3bitr4g 223	1 |- ( A e. V -> ( A e. { B , C , D } <-> ( A = B \/ A = C \/ A = D ) ) )

Syntax hints:   -> wi 4   <-> wb 105   \/ wo 713   \/ w3o 1001   = wceq 1395   e. wcel 2200   u. cun 3195   {csn 3666   {cpr 3667   {ctp 3668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-sn 3672  df-pr 3673  df-tp 3674

This theorem is referenced by:  eldiftp  3712  eltpi  3713  eltp  3714  tpid1g  3778  tpid2g  3780  zabsle1  15672  gausslemma2dlem0i  15730  2lgsoddprm  15786