Theorem eltp 3717

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 3714	. 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 1003   = wceq 1397   e. wcel 2202   _Vcvv 2802   {ctp 3671

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-tp 3677

This theorem is referenced by:  dftp2  3718  tpid1  3783  tpid2  3785  tpid3  3788