Theorem eltpg 3770

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 3751	. . 3 |- (A e. V -> (A e. {B, C} <-> (A = B \/ A = C)))
2		elsncg 3756	. . 3 |- (A e. V -> (A e. {D} <-> A = D))
3	1, 2	orbi12d 690	. 2 |- (A e. V -> ((A e. {B, C} \/ A e. {D}) <-> ((A = B \/ A = C) \/ A = D)))
4		df-tp 3744	. . . 4 |- {B, C, D} = ({B, C} u. {D})
5	4	eleq2i 2417	. . 3 |- (A e. {B, C, D} <-> A e. ({B, C} u. {D}))
6		elun 3221	. . 3 |- (A e. ({B, C} u. {D}) <-> (A e. {B, C} \/ A e. {D}))
7	5, 6	bitri 240	. 2 |- (A e. {B, C, D} <-> (A e. {B, C} \/ A e. {D}))
8		df-3or 935	. 2 |- ((A = B \/ A = C \/ A = D) <-> ((A = B \/ A = C) \/ A = D))
9	3, 7, 8	3bitr4g 279	1 |- (A e. V -> (A e. {B, C, D} <-> (A = B \/ A = C \/ A = D)))

Syntax hints:   -> wi 4   <-> wb 176   \/ wo 357   \/ w3o 933   = wceq 1642   e. wcel 1710   u. cun 3208  {csn 3738  {cpr 3739  {ctp 3740

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743  df-tp 3744

This theorem is referenced by:  eltpi  3771  eltp  3772