Theorem eltp 4019

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 4016	. 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 188   \/ w3o 985   = wceq 1446   e. wcel 1889   _Vcvv 3047   {ctp 3974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-v 3049  df-un 3411  df-sn 3971  df-pr 3973  df-tp 3975

This theorem is referenced by:  dftp2  4020  tpid1  4088  tpid2  4089  tpid3  4091  tpres  6122  bpoly3  14123  nb3graprlem1  25191  frgra3vlem1  25740  frgra3vlem2  25741  brtp  30401  sltsolem1  30569  nb3grprlem1  39464