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Theorem eltp 3638
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 3636 . 2  |-  ( A  e.  _V  ->  ( A  e.  { B ,  C ,  D }  <->  ( A  =  B  \/  A  =  C  \/  A  =  D )
) )
31, 2ax-mp 10 1  |-  ( A  e.  { B ,  C ,  D }  <->  ( A  =  B  \/  A  =  C  \/  A  =  D )
)
Colors of variables: wff set class
Syntax hints:    <-> wb 178    \/ w3o 938    = wceq 1619    e. wcel 1621   _Vcvv 2757   {ctp 3602
This theorem is referenced by:  dftp2  3639  tpid1  3699  tpid2  3700  tpid3  3702  brtp  23463  axsltsolem1  23676  bpoly3  24154  fnckle  25398
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-v 2759  df-un 3118  df-sn 3606  df-pr 3607  df-tp 3608
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