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

Theorem eltpg 3980
Description: Members of an unordered triple of classes. (Contributed by FL, 2-Feb-2014.) (Proof shortened by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
eltpg  |-  ( A  e.  V  ->  ( A  e.  { B ,  C ,  D }  <->  ( A  =  B  \/  A  =  C  \/  A  =  D )
) )

Proof of Theorem eltpg
StepHypRef Expression
1 elprg 3952 . . 3  |-  ( A  e.  V  ->  ( A  e.  { B ,  C }  <->  ( A  =  B  \/  A  =  C ) ) )
2 elsncg 3959 . . 3  |-  ( A  e.  V  ->  ( A  e.  { D } 
<->  A  =  D ) )
31, 2orbi12d 714 . 2  |-  ( A  e.  V  ->  (
( A  e.  { B ,  C }  \/  A  e.  { D } )  <->  ( ( A  =  B  \/  A  =  C )  \/  A  =  D
) ) )
4 df-tp 3941 . . . 4  |-  { B ,  C ,  D }  =  ( { B ,  C }  u.  { D } )
54eleq2i 2493 . . 3  |-  ( A  e.  { B ,  C ,  D }  <->  A  e.  ( { B ,  C }  u.  { D } ) )
6 elun 3544 . . 3  |-  ( A  e.  ( { B ,  C }  u.  { D } )  <->  ( A  e.  { B ,  C }  \/  A  e.  { D } ) )
75, 6bitri 252 . 2  |-  ( A  e.  { B ,  C ,  D }  <->  ( A  e.  { B ,  C }  \/  A  e.  { D } ) )
8 df-3or 983 . 2  |-  ( ( A  =  B  \/  A  =  C  \/  A  =  D )  <->  ( ( A  =  B  \/  A  =  C )  \/  A  =  D ) )
93, 7, 83bitr4g 291 1  |-  ( A  e.  V  ->  ( A  e.  { B ,  C ,  D }  <->  ( A  =  B  \/  A  =  C  \/  A  =  D )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    \/ wo 369    \/ w3o 981    = wceq 1437    e. wcel 1872    u. cun 3372   {csn 3936   {cpr 3938   {ctp 3940
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 2403
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 2410  df-cleq 2416  df-clel 2419  df-nfc 2553  df-v 3019  df-un 3379  df-sn 3937  df-pr 3939  df-tp 3941
This theorem is referenced by:  eldiftp  3981  eltpi  3982  eltp  3983  f1dom3fv3dif  6122  f1dom3el3dif  6123  lcmftp  14547  estrreslem2  15961  1cubr  23705  nb3graprlem1  25116  nb3grprlem1  39185
  Copyright terms: Public domain W3C validator