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Theorem tpid3 3742
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 2283 . . 3  |-  C  =  C
213mix3i 1129 . 2  |-  ( C  =  A  \/  C  =  B  \/  C  =  C )
3 tpid3.1 . . 3  |-  C  e. 
_V
43eltp 3678 . 2  |-  ( C  e.  { A ,  B ,  C }  <->  ( C  =  A  \/  C  =  B  \/  C  =  C )
)
52, 4mpbir 200 1  |-  C  e. 
{ A ,  B ,  C }
Colors of variables: wff set class
Syntax hints:    \/ w3o 933    = wceq 1623    e. wcel 1684   _Vcvv 2788   {ctp 3642
This theorem is referenced by:  ex-pss  20815  kur14lem7  23743  brtpid3  24077  pfsubkl  26047  rabren3dioph  26898
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-un 3157  df-sn 3646  df-pr 3647  df-tp 3648
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