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Theorem tpass 4092
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 3998 . 2  |-  { B ,  C ,  A }  =  ( { B ,  C }  u.  { A } )
2 tprot 4089 . 2  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
3 uncom 3607 . 2  |-  ( { A }  u.  { B ,  C }
)  =  ( { B ,  C }  u.  { A } )
41, 2, 33eqtr4i 2459 1  |-  { A ,  B ,  C }  =  ( { A }  u.  { B ,  C } )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437    u. cun 3431   {csn 3993   {cpr 3995   {ctp 3997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-v 3080  df-un 3438  df-sn 3994  df-pr 3996  df-tp 3998
This theorem is referenced by:  qdassr  4094  en3  7805  wuntp  9125  ex-pw  25721
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