Theorem eltpg 3683

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 3658	. . 3 |- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )
2		elsng 3653	. . 3 |- ( A e. V -> ( A e. { D } <-> A = D ) )
3	1, 2	orbi12d 795	. 2 |- ( A e. V -> ( ( A e. { B , C } \/ A e. { D } ) <-> ( ( A = B \/ A = C ) \/ A = D ) ) )
4		df-tp 3646	. . . 4 |- { B , C , D } = ( { B , C } u. { D } )
5	4	eleq2i 2273	. . 3 |- ( A e. { B , C , D } <-> A e. ( { B , C } u. { D } ) )
6		elun 3318	. . 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 982	. 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 710   \/ w3o 980   = wceq 1373   e. wcel 2177   u. cun 3168   {csn 3638   {cpr 3639   {ctp 3640

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3or 982  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-v 2775  df-un 3174  df-sn 3644  df-pr 3645  df-tp 3646

This theorem is referenced by:  eldiftp  3684  eltpi  3685  eltp  3686  tpid1g  3750  tpid2g  3752  zabsle1  15551  gausslemma2dlem0i  15609  2lgsoddprm  15665