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Theorem tpid1 3815
Description: One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Hypothesis
Ref Expression
tpid1.1  |-  A  e. 
_V
Assertion
Ref Expression
tpid1  |-  A  e. 
{ A ,  B ,  C }

Proof of Theorem tpid1
StepHypRef Expression
1 eqid 2358 . . 3  |-  A  =  A
213mix1i 1127 . 2  |-  ( A  =  A  \/  A  =  B  \/  A  =  C )
3 tpid1.1 . . 3  |-  A  e. 
_V
43eltp 3754 . 2  |-  ( A  e.  { A ,  B ,  C }  <->  ( A  =  A  \/  A  =  B  \/  A  =  C )
)
52, 4mpbir 200 1  |-  A  e. 
{ A ,  B ,  C }
Colors of variables: wff set class
Syntax hints:    \/ w3o 933    = wceq 1642    e. wcel 1710   _Vcvv 2864   {ctp 3718
This theorem is referenced by:  tpnz  3823  kur14lem7  24147  kur14lem9  24149  brtpid1  24479  rabren3dioph  26221  2pthonlem2  27736
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-v 2866  df-un 3233  df-sn 3722  df-pr 3723  df-tp 3724
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