ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  tpid2 Unicode version

Theorem tpid2 3512
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 2082 . . 3  |-  B  =  B
213mix2i 1112 . 2  |-  ( B  =  A  \/  B  =  B  \/  B  =  C )
3 tpid2.1 . . 3  |-  B  e. 
_V
43eltp 3448 . 2  |-  ( B  e.  { A ,  B ,  C }  <->  ( B  =  A  \/  B  =  B  \/  B  =  C )
)
52, 4mpbir 144 1  |-  B  e. 
{ A ,  B ,  C }
Colors of variables: wff set class
Syntax hints:    \/ w3o 919    = wceq 1285    e. wcel 1434   _Vcvv 2602   {ctp 3408
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3or 921  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-sn 3412  df-pr 3413  df-tp 3414
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator