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Theorem qdass 3846
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 3447 . 2  |-  ( ( { A ,  B }  u.  { C } )  u.  { D } )  =  ( { A ,  B }  u.  ( { C }  u.  { D } ) )
2 df-tp 3765 . . 3  |-  { A ,  B ,  C }  =  ( { A ,  B }  u.  { C } )
32uneq1i 3440 . 2  |-  ( { A ,  B ,  C }  u.  { D } )  =  ( ( { A ,  B }  u.  { C } )  u.  { D } )
4 df-pr 3764 . . 3  |-  { C ,  D }  =  ( { C }  u.  { D } )
54uneq2i 3441 . 2  |-  ( { A ,  B }  u.  { C ,  D } )  =  ( { A ,  B }  u.  ( { C }  u.  { D } ) )
61, 3, 53eqtr4ri 2418 1  |-  ( { A ,  B }  u.  { C ,  D } )  =  ( { A ,  B ,  C }  u.  { D } )
Colors of variables: wff set class
Syntax hints:    = wceq 1649    u. cun 3261   {csn 3757   {cpr 3758   {ctp 3759
This theorem is referenced by:  ex-pw  21585
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 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-v 2901  df-un 3268  df-pr 3764  df-tp 3765
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