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Theorem tpid3 4122
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 2423 . . 3  |-  C  =  C
213mix3i 1180 . 2  |-  ( C  =  A  \/  C  =  B  \/  C  =  C )
3 tpid3.1 . . 3  |-  C  e. 
_V
43eltp 4051 . 2  |-  ( C  e.  { A ,  B ,  C }  <->  ( C  =  A  \/  C  =  B  \/  C  =  C )
)
52, 4mpbir 213 1  |-  C  e. 
{ A ,  B ,  C }
Colors of variables: wff setvar class
Syntax hints:    \/ w3o 982    = wceq 1438    e. wcel 1873   _Vcvv 3085   {ctp 4008
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1664  ax-4 1677  ax-5 1753  ax-6 1799  ax-7 1844  ax-10 1892  ax-11 1897  ax-12 1910  ax-13 2058  ax-ext 2402
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-tru 1441  df-ex 1659  df-nf 1663  df-sb 1792  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-v 3087  df-un 3447  df-sn 4005  df-pr 4007  df-tp 4009
This theorem is referenced by:  2pthlem2  25330  usgra2adedgwlkonALT  25348  ex-pss  25882  sgnsf  28505  sgncl  29423  kur14lem7  29949  brtpid3  30369  rabren3dioph  35633  fourierdlem114  38030
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