Theorem tpid1 3694

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 2170	. . 3 |- A = A
2	1	3mix1i 1164	. 2 |- ( A = A \/ A = B \/ A = C )
3		tpid1.1	. . 3 |- A e. _V
4	3	eltp 3631	. 2 |- ( A e. { A , B , C } <-> ( A = A \/ A = B \/ A = C ) )
5	2, 4	mpbir 145	1 |- A e. { A , B , C }

Syntax hints:   \/ w3o 972   = wceq 1348   e. wcel 2141   _Vcvv 2730   {ctp 3585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3or 974  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-tp 3591

This theorem is referenced by:  tpnz  3708