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Theorem tpid3 3734
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 2193 . . 3  |-  C  =  C
213mix3i 1173 . 2  |-  ( C  =  A  \/  C  =  B  \/  C  =  C )
3 tpid3.1 . . 3  |-  C  e. 
_V
43eltp 3666 . 2  |-  ( C  e.  { A ,  B ,  C }  <->  ( C  =  A  \/  C  =  B  \/  C  =  C )
)
52, 4mpbir 146 1  |-  C  e. 
{ A ,  B ,  C }
Colors of variables: wff set class
Syntax hints:    \/ w3o 979    = wceq 1364    e. wcel 2164   _Vcvv 2760   {ctp 3620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-3or 981  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-sn 3624  df-pr 3625  df-tp 3626
This theorem is referenced by: (None)
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