Theorem tpid3 3749

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 2205	. . 3 |- C = C
2	1	3mix3i 1174	. 2 |- ( C = A \/ C = B \/ C = C )
3		tpid3.1	. . 3 |- C e. _V
4	3	eltp 3681	. 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 980   = wceq 1373   e. wcel 2176   _Vcvv 2772   {ctp 3635

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3or 982  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-sn 3639  df-pr 3640  df-tp 3641

This theorem is referenced by: (None)