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Theorem tpass 3738
Description: Split off the first element of an unordered triple. (Contributed by Mario Carneiro, 5-Jan-2016.)
Assertion
Ref Expression
tpass  |-  { A ,  B ,  C }  =  ( { A }  u.  { B ,  C } )

Proof of Theorem tpass
StepHypRef Expression
1 df-tp 3661 . 2  |-  { B ,  C ,  A }  =  ( { B ,  C }  u.  { A } )
2 tprot 3735 . 2  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
3 uncom 3332 . 2  |-  ( { A }  u.  { B ,  C }
)  =  ( { B ,  C }  u.  { A } )
41, 2, 33eqtr4i 2326 1  |-  { A ,  B ,  C }  =  ( { A }  u.  { B ,  C } )
Colors of variables: wff set class
Syntax hints:    = wceq 1632    u. cun 3163   {csn 3653   {cpr 3654   {ctp 3655
This theorem is referenced by:  qdassr  3740  en3  7111  wuntp  8349  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
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