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Theorem qdass 3903
Description: Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)
Assertion
Ref Expression
qdass  |-  ( { A ,  B }  u.  { C ,  D } )  =  ( { A ,  B ,  C }  u.  { D } )

Proof of Theorem qdass
StepHypRef Expression
1 unass 3504 . 2  |-  ( ( { A ,  B }  u.  { C } )  u.  { D } )  =  ( { A ,  B }  u.  ( { C }  u.  { D } ) )
2 df-tp 3822 . . 3  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
32uneq1i 3497 . 2  |-  ( { A ,  B ,  C }  u.  { D } )  =  ( ( { A ,  B }  u.  { C } )  u.  { D } )
4 df-pr 3821 . . 3  |-  { C ,  D }  =  ( { C }  u.  { D } )
54uneq2i 3498 . 2  |-  ( { A ,  B }  u.  { C ,  D } )  =  ( { A ,  B }  u.  ( { C }  u.  { D } ) )
61, 3, 53eqtr4ri 2467 1  |-  ( { A ,  B }  u.  { C ,  D } )  =  ( { A ,  B ,  C }  u.  { D } )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    u. cun 3318   {csn 3814   {cpr 3815   {ctp 3816
This theorem is referenced by:  ex-pw  21737
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 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-un 3325  df-pr 3821  df-tp 3822
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