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

Theorem qdassr 3740
Description: Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)
Assertion
Ref Expression
qdassr  |-  ( { A ,  B }  u.  { C ,  D } )  =  ( { A }  u.  { B ,  C ,  D } )

Proof of Theorem qdassr
StepHypRef Expression
1 unass 3345 . 2  |-  ( ( { A }  u.  { B } )  u. 
{ C ,  D } )  =  ( { A }  u.  ( { B }  u.  { C ,  D }
) )
2 df-pr 3660 . . 3  |-  { A ,  B }  =  ( { A }  u.  { B } )
32uneq1i 3338 . 2  |-  ( { A ,  B }  u.  { C ,  D } )  =  ( ( { A }  u.  { B } )  u.  { C ,  D } )
4 tpass 3738 . . 3  |-  { B ,  C ,  D }  =  ( { B }  u.  { C ,  D } )
54uneq2i 3339 . 2  |-  ( { A }  u.  { B ,  C ,  D } )  =  ( { A }  u.  ( { B }  u.  { C ,  D }
) )
61, 3, 53eqtr4i 2326 1  |-  ( { A ,  B }  u.  { C ,  D } )  =  ( { A }  u.  { B ,  C ,  D } )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    u. cun 3163   {csn 3653   {cpr 3654   {ctp 3655
This theorem is referenced by:  en4  7112  ex-pw  20832
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-v 2803  df-un 3170  df-sn 3659  df-pr 3660  df-tp 3661
  Copyright terms: Public domain W3C validator