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Theorem tpid3 4119
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 2429 . . 3  |-  C  =  C
213mix3i 1179 . 2  |-  ( C  =  A  \/  C  =  B  \/  C  =  C )
3 tpid3.1 . . 3  |-  C  e. 
_V
43eltp 4048 . 2  |-  ( C  e.  { A ,  B ,  C }  <->  ( C  =  A  \/  C  =  B  \/  C  =  C )
)
52, 4mpbir 212 1  |-  C  e. 
{ A ,  B ,  C }
Colors of variables: wff setvar class
Syntax hints:    \/ w3o 981    = wceq 1437    e. wcel 1870   _Vcvv 3087   {ctp 4006
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-v 3089  df-un 3447  df-sn 4003  df-pr 4005  df-tp 4007
This theorem is referenced by:  2pthlem2  25171  usgra2adedgwlkonALT  25189  ex-pss  25723  sgnsf  28330  sgncl  29197  kur14lem7  29723  brtpid3  30143  rabren3dioph  35367  fourierdlem114  37652
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