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Theorem tpid1 3877
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 2404 . . 3  |-  A  =  A
213mix1i 1129 . 2  |-  ( A  =  A  \/  A  =  B  \/  A  =  C )
3 tpid1.1 . . 3  |-  A  e. 
_V
43eltp 3813 . 2  |-  ( A  e.  { A ,  B ,  C }  <->  ( A  =  A  \/  A  =  B  \/  A  =  C )
)
52, 4mpbir 201 1  |-  A  e. 
{ A ,  B ,  C }
Colors of variables: wff set class
Syntax hints:    \/ w3o 935    = wceq 1649    e. wcel 1721   _Vcvv 2916   {ctp 3776
This theorem is referenced by:  tpnz  3885  2pthlem2  21549  kur14lem7  24851  kur14lem9  24853  brtpid1  25131  rabren3dioph  26766
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-v 2918  df-un 3285  df-sn 3780  df-pr 3781  df-tp 3782
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