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Theorem tpass 4645
 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} ∪ {𝐵, 𝐶})

Proof of Theorem tpass
StepHypRef Expression
1 df-tp 4527 . 2 {𝐵, 𝐶, 𝐴} = ({𝐵, 𝐶} ∪ {𝐴})
2 tprot 4642 . 2 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
3 uncom 4058 . 2 ({𝐴} ∪ {𝐵, 𝐶}) = ({𝐵, 𝐶} ∪ {𝐴})
41, 2, 33eqtr4i 2791 1 {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶})
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∪ cun 3856  {csn 4522  {cpr 4524  {ctp 4526 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-un 3863  df-sn 4523  df-pr 4525  df-tp 4527 This theorem is referenced by:  qdassr  4647  en3  8791  wuntp  10171  symgvalstruct  18592  ex-pw  28313
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