Theorem eltp 3756

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 3753	. 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 1008   = wceq 1402   e. wcel 2209   _Vcvv 2821   {ctp 3710

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715  df-tp 3716

This theorem is referenced by:  dftp2  3757  tpid1  3822  tpid2  3824  tpid3  3827