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Theorem tpass 3589
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 3505 . 2 {𝐵, 𝐶, 𝐴} = ({𝐵, 𝐶} ∪ {𝐴})
2 tprot 3586 . 2 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
3 uncom 3190 . 2 ({𝐴} ∪ {𝐵, 𝐶}) = ({𝐵, 𝐶} ∪ {𝐴})
41, 2, 33eqtr4i 2148 1 {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶})
Colors of variables: wff set class
Syntax hints:   = wceq 1316  cun 3039  {csn 3497  {cpr 3498  {ctp 3499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3or 948  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-sn 3503  df-pr 3504  df-tp 3505
This theorem is referenced by:  qdassr  3591
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