Theorem eltp 3074

Description: A member of an unordered triple of classes is one of them. Special case of Exercise 1 of [TakeutiZaring] p. 17.

Hypothesis

Ref	Expression
eltp.1	|- A e. _V

Assertion

Ref	Expression
eltp	|- (A e. {B, C, D} <-> (A = B \/ A = C \/ A = D))

Proof of Theorem eltp

Step	Hyp	Ref	Expression
1		df-tp 3052	. . . 4 |- {B, C, D} = ({B, C} u. {D})
2	1	eleq2i 1961	. . 3 |- (A e. {B, C, D} <-> A e. ({B, C} u. {D}))
3		elun 2741	. . 3 |- (A e. ({B, C} u. {D}) <-> (A e. {B, C} \/ A e. {D}))
4		eltp.1	. . . . 5 |- A e. _V
5	4	elpr 3061	. . . 4 |- (A e. {B, C} <-> (A = B \/ A = C))
6	4	elsnc 3065	. . . 4 |- (A e. {D} <-> A = D)
7	5, 6	orbi12i 277	. . 3 |- ((A e. {B, C} \/ A e. {D}) <-> ((A = B \/ A = C) \/ A = D))
8	2, 3, 7	3bitri 194	. 2 |- (A e. {B, C, D} <-> ((A = B \/ A = C) \/ A = D))
9		df-3or 859	. 2 |- ((A = B \/ A = C \/ A = D) <-> ((A = B \/ A = C) \/ A = D))
10	8, 9	bitr4i 193	1 |- (A e. {B, C, D} <-> (A = B \/ A = C \/ A = D))

Syntax hints:   <-> wb 163   \/ wo 239   \/ w3o 857   = wceq 1298   e. wcel 1300  _Vcvv 2292   u. cun 2591  {csn 3044  {cpr 3045  {ctp 3051

This theorem is referenced by:  dftp2 3075  raltp 3083  tpid1 3111  tpid2 3113  tpid3 3116  tpssOLD 3146  fr3nr 3859  brtp 13830  axsltsolem1 14006  stb3xpl 16743

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-un 2600  df-sn 3049  df-pr 3050  df-tp 3052

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