Theorem tpass 3488

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

Step	Hyp	Ref	Expression
1		df-tp 3406	. 2 ⊢ {𝐵, 𝐶, 𝐴} = ({𝐵, 𝐶} ∪ {𝐴})
2		tprot 3485	. 2 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
3		uncom 3116	. 2 ⊢ ({𝐴} ∪ {𝐵, 𝐶}) = ({𝐵, 𝐶} ∪ {𝐴})
4	1, 2, 3	3eqtr4i 2111	1 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴} ∪ {𝐵, 𝐶})

Syntax hints:   = wceq 1284   ∪ cun 2971  {csn 3398  {cpr 3399  {ctp 3400

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3or 920  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-sn 3404  df-pr 3405  df-tp 3406

This theorem is referenced by:  qdassr  3490