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Theorem tpid2 4050
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
StepHypRef Expression
1 eqid 2422 . . 3  |-  B  =  B
213mix2i 1178 . 2  |-  ( B  =  A  \/  B  =  B  \/  B  =  C )
3 tpid2.1 . . 3  |-  B  e. 
_V
43eltp 3981 . 2  |-  ( B  e.  { A ,  B ,  C }  <->  ( B  =  A  \/  B  =  B  \/  B  =  C )
)
52, 4mpbir 212 1  |-  B  e. 
{ A ,  B ,  C }
Colors of variables: wff setvar class
Syntax hints:    \/ w3o 981    = wceq 1437    e. wcel 1872   _Vcvv 3016   {ctp 3938
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-v 3018  df-un 3377  df-sn 3935  df-pr 3937  df-tp 3939
This theorem is referenced by:  2pthlem1  25260  2pthlem2  25261  el2wlkonotlem  25525  sgnsf  28436  sgncl  29354  signsw0glem  29387  signsw0g  29390  signswmnd  29391  signswrid  29392  kur14lem7  29880  brtpid2  30299  rabren3dioph  35564  fourierdlem102  37949  fourierdlem114  37961  etransclem48OLD  38024  etransclem48  38025
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