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Theorem tpass 4131
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 4038 . 2  |-  { B ,  C ,  A }  =  ( { B ,  C }  u.  { A } )
2 tprot 4128 . 2  |-  { A ,  B ,  C }  =  { B ,  C ,  A }
3 uncom 3653 . 2  |-  ( { A }  u.  { B ,  C }
)  =  ( { B ,  C }  u.  { A } )
41, 2, 33eqtr4i 2506 1  |-  { A ,  B ,  C }  =  ( { A }  u.  { B ,  C } )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1379    u. cun 3479   {csn 4033   {cpr 4035   {ctp 4037
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3120  df-un 3486  df-sn 4034  df-pr 4036  df-tp 4038
This theorem is referenced by:  qdassr  4133  en3  7769  wuntp  9101  ex-pw  24974
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