Theorem tpid2 3482

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
tpid2.1	|- B e. _V

Assertion

Ref	Expression
tpid2	|- B e. { A , B , C }

Proof of Theorem tpid2

Step	Hyp	Ref	Expression
1		eqid 2040	. . 3 |- B = B
2	1	3mix2i 1077	. 2 |- ( B = A \/ B = B \/ B = C )
3		tpid2.1	. . 3 |- B e. _V
4	3	eltp 3418	. 2 |- ( B e. { A , B , C } <-> ( B = A \/ B = B \/ B = C ) )
5	2, 4	mpbir 134	1 |- B e. { A , B , C }

Syntax hints:   \/ w3o 884   = wceq 1243   e. wcel 1393   _Vcvv 2557   {ctp 3377

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3or 886  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-sn 3381  df-pr 3382  df-tp 3383

This theorem is referenced by: (None)