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Theorem qdassr 3896
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 3496 . 2  |-  ( ( { A }  u.  { B } )  u. 
{ C ,  D } )  =  ( { A }  u.  ( { B }  u.  { C ,  D }
) )
2 df-pr 3813 . . 3  |-  { A ,  B }  =  ( { A }  u.  { B } )
32uneq1i 3489 . 2  |-  ( { A ,  B }  u.  { C ,  D } )  =  ( ( { A }  u.  { B } )  u.  { C ,  D } )
4 tpass 3894 . . 3  |-  { B ,  C ,  D }  =  ( { B }  u.  { C ,  D } )
54uneq2i 3490 . 2  |-  ( { A }  u.  { B ,  C ,  D } )  =  ( { A }  u.  ( { B }  u.  { C ,  D }
) )
61, 3, 53eqtr4i 2465 1  |-  ( { A ,  B }  u.  { C ,  D } )  =  ( { A }  u.  { B ,  C ,  D } )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    u. cun 3310   {csn 3806   {cpr 3807   {ctp 3808
This theorem is referenced by:  en4  7338  ex-pw  21727
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-un 3317  df-sn 3812  df-pr 3813  df-tp 3814
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