Theorem eltp 2435

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 2411	. . . 4 |- {B, C, D} = ({B, C} u. {D})
2	1	eleq2i 1535	. . 3 |- (A e. {B, C, D} <-> A e. ({B, C} u. {D}))
3		elun 2169	. . 3 |- (A e. ({B, C} u. {D}) <-> (A e. {B, C} \/ A e. {D}))
4		eltp.1	. . . . 5 |- A e. V
5	4	elpr 2420	. . . 4 |- (A e. {B, C} <-> (A = B \/ A = C))
6	4	elsnc 2427	. . . 4 |- (A e. {D} <-> A = D)
7	5, 6	orbi12i 257	. . 3 |- ((A e. {B, C} \/ A e. {D}) <-> ((A = B \/ A = C) \/ A = D))
8	2, 3, 7	3bitr 177	. 2 |- (A e. {B, C, D} <-> ((A = B \/ A = C) \/ A = D))
9		df-3or 775	. 2 |- ((A = B \/ A = C \/ A = D) <-> ((A = B \/ A = C) \/ A = D))
10	8, 9	bitr4 176	1 |- (A e. {B, C, D} <-> (A = B \/ A = C \/ A = D))

Syntax hints:   <-> wb 146   \/ wo 222   \/ w3o 773   = wceq 954   e. wcel 956  Vcvv 1807   u. cun 2041  {csn 2405  {cpr 2406  {ctp 2410

This theorem is referenced by:  dftp2 2436  tpss 2472  fr3nr 2921

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-un 2046  df-sn 2408  df-pr 2409  df-tp 2411

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