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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
StepHypRef 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 )
) )
31, 2ax-mp 5 1  |-  ( A  e.  { B ,  C ,  D }  <->  ( A  =  B  \/  A  =  C  \/  A  =  D )
)
Colors of variables: wff setvar class
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
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