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Theorem tpass 3902
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 3822 . 2  |-  { B ,  C ,  A }  =  ( { B ,  C }  u.  { A } )
2 tprot 3899 . 2  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
3 uncom 3491 . 2  |-  ( { A }  u.  { B ,  C }
)  =  ( { B ,  C }  u.  { A } )
41, 2, 33eqtr4i 2466 1  |-  { A ,  B ,  C }  =  ( { A }  u.  { B ,  C } )
Colors of variables: wff set class
Syntax hints:    = wceq 1652    u. cun 3318   {csn 3814   {cpr 3815   {ctp 3816
This theorem is referenced by:  qdassr  3904  en3  7345  wuntp  8586  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-3or 937  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-sn 3820  df-pr 3821  df-tp 3822
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