MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tpid3 Structured version   Unicode version

Theorem tpid3 4132
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
tpid3.1  |-  C  e. 
_V
Assertion
Ref Expression
tpid3  |-  C  e. 
{ A ,  B ,  C }

Proof of Theorem tpid3
StepHypRef Expression
1 eqid 2454 . . 3  |-  C  =  C
213mix3i 1168 . 2  |-  ( C  =  A  \/  C  =  B  \/  C  =  C )
3 tpid3.1 . . 3  |-  C  e. 
_V
43eltp 4061 . 2  |-  ( C  e.  { A ,  B ,  C }  <->  ( C  =  A  \/  C  =  B  \/  C  =  C )
)
52, 4mpbir 209 1  |-  C  e. 
{ A ,  B ,  C }
Colors of variables: wff setvar class
Syntax hints:    \/ w3o 970    = wceq 1398    e. wcel 1823   _Vcvv 3106   {ctp 4020
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3108  df-un 3466  df-sn 4017  df-pr 4019  df-tp 4021
This theorem is referenced by:  2pthlem2  24800  usgra2adedgwlkonALT  24818  ex-pss  25351  sgnsf  27953  sgncl  28741  kur14lem7  28920  brtpid3  29341  rabren3dioph  30988  fourierdlem114  32242
  Copyright terms: Public domain W3C validator