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Theorem tpid3 4092
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
tpid3.1  |-  C  e. 
_V
Assertion
Ref Expression
tpid3  |-  C  e. 
{ A ,  B ,  C }

Proof of Theorem tpid3
StepHypRef Expression
1 eqid 2451 . . 3  |-  C  =  C
213mix3i 1162 . 2  |-  ( C  =  A  \/  C  =  B  \/  C  =  C )
3 tpid3.1 . . 3  |-  C  e. 
_V
43eltp 4022 . 2  |-  ( C  e.  { A ,  B ,  C }  <->  ( C  =  A  \/  C  =  B  \/  C  =  C )
)
52, 4mpbir 209 1  |-  C  e. 
{ A ,  B ,  C }
Colors of variables: wff setvar class
Syntax hints:    \/ w3o 964    = wceq 1370    e. wcel 1758   _Vcvv 3071   {ctp 3982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-v 3073  df-un 3434  df-sn 3979  df-pr 3981  df-tp 3983
This theorem is referenced by:  2pthlem2  23640  ex-pss  23780  sgnsf  26330  sgncl  27058  kur14lem7  27237  brtpid3  27516  rabren3dioph  29295
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