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

Theorem eltpi 4013
Description: A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
eltpi  |-  ( A  e.  { B ,  C ,  D }  ->  ( A  =  B  \/  A  =  C  \/  A  =  D ) )

Proof of Theorem eltpi
StepHypRef Expression
1 eltpg 4011 . 2  |-  ( A  e.  { B ,  C ,  D }  ->  ( A  e.  { B ,  C ,  D }  <->  ( A  =  B  \/  A  =  C  \/  A  =  D ) ) )
21ibi 241 1  |-  ( A  e.  { B ,  C ,  D }  ->  ( A  =  B  \/  A  =  C  \/  A  =  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ w3o 971    = wceq 1403    e. wcel 1840   {ctp 3973
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-v 3058  df-un 3416  df-sn 3970  df-pr 3972  df-tp 3974
This theorem is referenced by:  perfectlem2  23776  sgnmulsgn  28875  sgnmulsgp  28876  kur14lem7  29385  perfectALTVlem2  37759
  Copyright terms: Public domain W3C validator