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Theorem tpass 4759
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 4635 . 2 {𝐵, 𝐶, 𝐴} = ({𝐵, 𝐶} ∪ {𝐴})
2 tprot 4756 . 2 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
3 uncom 4152 . 2 ({𝐴} ∪ {𝐵, 𝐶}) = ({𝐵, 𝐶} ∪ {𝐴})
41, 2, 33eqtr4i 2765 1 {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶})
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  cun 3945  {csn 4630  {cpr 4632  {ctp 4634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-v 3473  df-un 3952  df-sn 4631  df-pr 4633  df-tp 4635
This theorem is referenced by:  qdassr  4761  en3  9311  wuntp  10740  symgvalstruct  19356  symgvalstructOLD  19357  ex-pw  30257
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