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Theorem tpass 4595
 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 4477 . 2 {𝐵, 𝐶, 𝐴} = ({𝐵, 𝐶} ∪ {𝐴})
2 tprot 4592 . 2 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
3 uncom 4050 . 2 ({𝐴} ∪ {𝐵, 𝐶}) = ({𝐵, 𝐶} ∪ {𝐴})
41, 2, 33eqtr4i 2829 1 {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶})
 Colors of variables: wff setvar class Syntax hints:   = wceq 1522   ∪ cun 3857  {csn 4472  {cpr 4474  {ctp 4476 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-ext 2769 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-v 3439  df-un 3864  df-sn 4473  df-pr 4475  df-tp 4477 This theorem is referenced by:  qdassr  4597  en3  8601  wuntp  9979  ex-pw  27900
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