Theorem eltp 3691

Description: A member of an unordered triple of classes is one of them. Special case of Exercise 1 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Apr-1994.) (Revised by Mario Carneiro, 11-Feb-2015.)

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		eltp.1	. 2 |- A e. _V
2		eltpg 3688	. 2 |- ( A e. _V -> ( A e. { B , C , D } <-> ( A = B \/ A = C \/ A = D ) ) )
3	1, 2	ax-mp 5	1 |- ( A e. { B , C , D } <-> ( A = B \/ A = C \/ A = D ) )

Syntax hints:   <-> wb 105   \/ w3o 980   = wceq 1373   e. wcel 2178   _Vcvv 2776   {ctp 3645

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 2189

This theorem depends on definitions:  df-bi 117  df-3or 982  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-tp 3651

This theorem is referenced by:  dftp2  3692  tpid1  3754  tpid2  3756  tpid3  3759