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Theorem eltp 3845
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 3843 . 2  |-  ( A  e.  _V  ->  ( A  e.  { B ,  C ,  D }  <->  ( A  =  B  \/  A  =  C  \/  A  =  D )
) )
31, 2ax-mp 8 1  |-  ( A  e.  { B ,  C ,  D }  <->  ( A  =  B  \/  A  =  C  \/  A  =  D )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    \/ w3o 935    = wceq 1652    e. wcel 1725   _Vcvv 2948   {ctp 3808
This theorem is referenced by:  dftp2  3846  tpid1  3909  tpid2  3910  tpid3  3912  nb3graprlem1  21448  brtp  25361  sltsolem1  25577  bpoly3  26052  frgra3vlem1  28248  frgra3vlem2  28249
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-un 3317  df-sn 3812  df-pr 3813  df-tp 3814
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