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Theorem eltp 4008
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 4005 . 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 189    \/ w3o 1006    = wceq 1452    e. wcel 1904   _Vcvv 3031   {ctp 3963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-v 3033  df-un 3395  df-sn 3960  df-pr 3962  df-tp 3964
This theorem is referenced by:  dftp2  4009  tpid1  4076  tpid2  4077  tpid3  4079  tpres  6133  fntpb  6140  bpoly3  14188  nb3graprlem1  25258  frgra3vlem1  25807  frgra3vlem2  25808  brtp  30460  sltsolem1  30628  nb3grprlem1  39618
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