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Theorem qdassr 3849
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 3449 . 2  |-  ( ( { A }  u.  { B } )  u. 
{ C ,  D } )  =  ( { A }  u.  ( { B }  u.  { C ,  D }
) )
2 df-pr 3766 . . 3  |-  { A ,  B }  =  ( { A }  u.  { B } )
32uneq1i 3442 . 2  |-  ( { A ,  B }  u.  { C ,  D } )  =  ( ( { A }  u.  { B } )  u.  { C ,  D } )
4 tpass 3847 . . 3  |-  { B ,  C ,  D }  =  ( { B }  u.  { C ,  D } )
54uneq2i 3443 . 2  |-  ( { A }  u.  { B ,  C ,  D } )  =  ( { A }  u.  ( { B }  u.  { C ,  D }
) )
61, 3, 53eqtr4i 2419 1  |-  ( { A ,  B }  u.  { C ,  D } )  =  ( { A }  u.  { B ,  C ,  D } )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    u. cun 3263   {csn 3759   {cpr 3760   {ctp 3761
This theorem is referenced by:  en4  7284  ex-pw  21587
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-v 2903  df-un 3270  df-sn 3765  df-pr 3766  df-tp 3767
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