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Theorem tpid1 4086
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 2451 . . 3  |-  A  =  A
213mix1i 1160 . 2  |-  ( A  =  A  \/  A  =  B  \/  A  =  C )
3 tpid1.1 . . 3  |-  A  e. 
_V
43eltp 4019 . 2  |-  ( A  e.  { A ,  B ,  C }  <->  ( A  =  A  \/  A  =  B  \/  A  =  C )
)
52, 4mpbir 209 1  |-  A  e. 
{ A ,  B ,  C }
Colors of variables: wff setvar class
Syntax hints:    \/ w3o 964    = wceq 1370    e. wcel 1758   _Vcvv 3068   {ctp 3979
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 3070  df-un 3431  df-sn 3976  df-pr 3978  df-tp 3980
This theorem is referenced by:  tpnz  4094  2pthlem2  23630  sgnsf  26326  sgncl  27055  kur14lem7  27234  kur14lem9  27236  brtpid1  27511  rabren3dioph  29292
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