Theorem tpid1 3754

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
tpid1.1	|- A e. _V

Assertion

Ref	Expression
tpid1	|- A e. { A , B , C }

Proof of Theorem tpid1

Step	Hyp	Ref	Expression
1		eqid 2207	. . 3 |- A = A
2	1	3mix1i 1172	. 2 |- ( A = A \/ A = B \/ A = C )
3		tpid1.1	. . 3 |- A e. _V
4	3	eltp 3691	. 2 |- ( A e. { A , B , C } <-> ( A = A \/ A = B \/ A = C ) )
5	2, 4	mpbir 146	1 |- A e. { A , B , C }

Syntax hints:   \/ w3o 980   = wceq 1373   e. wcel 2178   _Vcvv 2776   {ctp 3645

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3or 982  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-tp 3651

This theorem is referenced by:  tpnz  3769