Theorem eltp 4043

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 4040	. 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 982   = wceq 1438   e. wcel 1869   _Vcvv 3082   {ctp 4001

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-v 3084  df-un 3442  df-sn 3998  df-pr 4000  df-tp 4002

This theorem is referenced by:  dftp2  4044  tpid1  4111  tpid2  4112  tpid3  4114  tpres  6130  bpoly3  14104  nb3graprlem1  25171  frgra3vlem1  25720  frgra3vlem2  25721  brtp  30390  sltsolem1  30556