Theorem tpid3 3730

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

Step	Hyp	Ref	Expression
1		eqid 2189	. . 3 |- C = C
2	1	3mix3i 1173	. 2 |- ( C = A \/ C = B \/ C = C )
3		tpid3.1	. . 3 |- C e. _V
4	3	eltp 3662	. 2 |- ( C e. { A , B , C } <-> ( C = A \/ C = B \/ C = C ) )
5	2, 4	mpbir 146	1 |- C e. { A , B , C }

Syntax hints:   \/ w3o 979   = wceq 1364   e. wcel 2160   _Vcvv 2756   {ctp 3616

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 2171

This theorem depends on definitions:  df-bi 117  df-3or 981  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2758  df-un 3152  df-sn 3620  df-pr 3621  df-tp 3622

This theorem is referenced by: (None)