Theorem tpid3 3827

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 2238	. . 3 |- C = C
2	1	3mix3i 1202	. 2 |- ( C = A \/ C = B \/ C = C )
3		tpid3.1	. . 3 |- C e. _V
4	3	eltp 3756	. 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 1008   = wceq 1402   e. wcel 2209   _Vcvv 2821   {ctp 3710

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715  df-tp 3716

This theorem is referenced by: (None)