Theorem tpid1 3743

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 2204	. . 3 |- A = A
2	1	3mix1i 1171	. 2 |- ( A = A \/ A = B \/ A = C )
3		tpid1.1	. . 3 |- A e. _V
4	3	eltp 3680	. 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 979   = wceq 1372   e. wcel 2175   _Vcvv 2771   {ctp 3634

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3or 981  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-sn 3638  df-pr 3639  df-tp 3640

This theorem is referenced by:  tpnz  3757