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Theorem tpass 4070
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 3973 . 2  |-  { B ,  C ,  A }  =  ( { B ,  C }  u.  { A } )
2 tprot 4067 . 2  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
3 uncom 3578 . 2  |-  ( { A }  u.  { B ,  C }
)  =  ( { B ,  C }  u.  { A } )
41, 2, 33eqtr4i 2483 1  |-  { A ,  B ,  C }  =  ( { A }  u.  { B ,  C } )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1444    u. cun 3402   {csn 3968   {cpr 3970   {ctp 3972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3047  df-un 3409  df-sn 3969  df-pr 3971  df-tp 3973
This theorem is referenced by:  qdassr  4072  en3  7808  wuntp  9136  ex-pw  25879
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