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Theorem tpass 4442
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 4339 . 2 {𝐵, 𝐶, 𝐴} = ({𝐵, 𝐶} ∪ {𝐴})
2 tprot 4439 . 2 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
3 uncom 3919 . 2 ({𝐴} ∪ {𝐵, 𝐶}) = ({𝐵, 𝐶} ∪ {𝐴})
41, 2, 33eqtr4i 2797 1 {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶})
Colors of variables: wff setvar class
Syntax hints:   = wceq 1652  cun 3730  {csn 4334  {cpr 4336  {ctp 4338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-un 3737  df-sn 4335  df-pr 4337  df-tp 4339
This theorem is referenced by:  qdassr  4444  en3  8404  wuntp  9786  ex-pw  27680
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