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Theorem tpid1 4113
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 2422 . . 3  |-  A  =  A
213mix1i 1177 . 2  |-  ( A  =  A  \/  A  =  B  \/  A  =  C )
3 tpid1.1 . . 3  |-  A  e. 
_V
43eltp 4045 . 2  |-  ( A  e.  { A ,  B ,  C }  <->  ( A  =  A  \/  A  =  B  \/  A  =  C )
)
52, 4mpbir 212 1  |-  A  e. 
{ A ,  B ,  C }
Colors of variables: wff setvar class
Syntax hints:    \/ w3o 981    = wceq 1437    e. wcel 1872   _Vcvv 3080   {ctp 4002
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-v 3082  df-un 3441  df-sn 3999  df-pr 4001  df-tp 4003
This theorem is referenced by:  tpnz  4121  2pthlem2  25324  usgra2adedgwlkonALT  25342  sgnsf  28499  sgncl  29417  kur14lem7  29943  kur14lem9  29945  brtpid1  30361  rabren3dioph  35627  fourierdlem102  38012  fourierdlem114  38024  etransclem48OLD  38087  etransclem48  38088
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