Theorem tpid2 4304

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 2622	. . 3 |- B = B
2	1	3mix2i 1234	. 2 |- ( B = A \/ B = B \/ B = C )
3		tpid2.1	. . 3 |- B e. _V
4	3	eltp 4230	. 2 |- ( B e. { A , B , C } <-> ( B = A \/ B = B \/ B = C ) )
5	2, 4	mpbir 221	1 |- B e. { A , B , C }

Syntax hints:   \/ w3o 1036   = wceq 1483   e. wcel 1990   _Vcvv 3200   {ctp 4181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-sn 4178  df-pr 4180  df-tp 4182

This theorem is referenced by:  wrdl3s3  13705  wwlks2onv  26847  elwwlks2ons3  26848  umgrwwlks2on  26850  sgnsf  29729  sgncl  30600  signsw0glem  30630  signsw0g  30633  signswmnd  30634  signswrid  30635  prodfzo03  30681  circlevma  30720  circlemethhgt  30721  hgt750lemg  30732  hgt750lemb  30734  hgt750lema  30735  hgt750leme  30736  tgoldbachgtde  30738  tgoldbachgt  30741  kur14lem7  31194  brtpid2  31603  rabren3dioph  37379  fourierdlem102  40425  fourierdlem114  40437  etransclem48  40499