Theorem eltpg 3667

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 3642	. . 3 |- ( A e. V -> ( A e. { B , C } <-> ( A = B \/ A = C ) ) )
2		elsng 3637	. . 3 |- ( A e. V -> ( A e. { D } <-> A = D ) )
3	1, 2	orbi12d 794	. 2 |- ( A e. V -> ( ( A e. { B , C } \/ A e. { D } ) <-> ( ( A = B \/ A = C ) \/ A = D ) ) )
4		df-tp 3630	. . . 4 |- { B , C , D } = ( { B , C } u. { D } )
5	4	eleq2i 2263	. . 3 |- ( A e. { B , C , D } <-> A e. ( { B , C } u. { D } ) )
6		elun 3304	. . 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 981	. 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 709   \/ w3o 979   = wceq 1364   e. wcel 2167   u. cun 3155   {csn 3622   {cpr 3623   {ctp 3624

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3or 981  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-sn 3628  df-pr 3629  df-tp 3630

This theorem is referenced by:  eldiftp  3668  eltpi  3669  eltp  3670  tpid1g  3734  tpid2g  3736  zabsle1  15240  gausslemma2dlem0i  15298  2lgsoddprm  15354